##
**Unsolved problems in number theory.**
*(English)*
Zbl 0474.10001

Problem Books in Mathematics. Unsolved Problems in Intuitive Mathematics, Vol. 1. New York - Heidelberg - Berlin: Springer-Verlag. XVIII, 161 p., 17 Figs. DM 42.00; $ 19.60 (1981).

“Mathematics is kept alive \(\dots\) by the appearence of a succession of unsolved problems, both from within mathematics itself and from the increasing number of disciplines where it is applied. Mathematics often owes more to those who ask questions than to those who answer then”. Getreu diesem Zitat aus dem Vorwort des vorliegenden, Paul Erdős gewidmeten Buches gibt der Verfasser eine umfangreiche Sammlung von Problemen aus dem Bereich der Zahlentheorie, die zu weiterer Beschäftigung mit zahlentheoretischen Fragestellungen anregen sollen. Zusätzlich gibt der Verfasser Hintergrundinformationen, d.h. eine Übersicht über bekannte Ergebnisse und ausführliche Literaturverweise nach jedem Problem, die gelegentlich 40 und mehr Zitate umfassen.

Die Probleme stammen aus folgenden Teilgebieten der Zahlentheorie:

A) Primzahlen (Primzahlen spezialler Gestalt, Primzahlen in arithmetischen Progressionen, Lücken zwischen Primzahlen, Zwillinge, Carmichael Zahlen, Gauß’sche Primzahlen, Formeln für \(p_n\) und andere).

B) Teilbarkeit (perfekte Zahlen, befreundete Zahlen, aliquot sequences, Lösungen von \(\sigma(n)= \sigma(n+1)\), ist \(\sum \sigma_k(n)/n!\) irrational, Cullen’sche Zahlen, Teilbarkeitseigenschaften von Fakultät und Binomialkoeffizienten, Carmicheal’s Vermutung, und viele andere).

C) Additive Zahlentheorie (Scholz’sche Vermutung, für welche \(N\) ist \(N= \sum a_i x_i\) unlörbar?, Folgen \(\{a_i\}\) für die alle \(a_i+a_j\) verschieden sind, perfekte Differenzmengen, Fragen über additive Basen, ‘maximal sum-free sets”, Damen auf dem Schachbrett, Darstellung durch vier teilerfremde Quadrate, u.a.).

D) Diophantische Gleichungen (im Umfeld der Fermat’schen Vermutung, spezielle diophantische Gleichungen, Catalan’s Vermutung, Darstellung durch Stammbrüche, und viele andere).

E) Folgen ganzer Zahlen (dünne Folgen \(\{a_i\}\), so daß \(p+a_i=n\) für große \(n\) lösbar wird, Konvergenz von \(\sum(-1)^n\cdot n/p_n\), Umfeld des van der Waerden’schen Satzes über arithmetische Progressionen, u.v.a.).

F) Vermischte Probleme.

Ein ausführliches Namens- und Sachverzeichnis beschließen das Buch. Der Verfasser hat dankenswerterweise eine umfangreiche Liste (20 pp.) von Errata und Addenda zu diesem Buch zusamemngestellt, die u.a. weitere Literaturverweise und weitere Ergebnisse numerischer Rechnungen enthalten.

Die Probleme stammen aus folgenden Teilgebieten der Zahlentheorie:

A) Primzahlen (Primzahlen spezialler Gestalt, Primzahlen in arithmetischen Progressionen, Lücken zwischen Primzahlen, Zwillinge, Carmichael Zahlen, Gauß’sche Primzahlen, Formeln für \(p_n\) und andere).

B) Teilbarkeit (perfekte Zahlen, befreundete Zahlen, aliquot sequences, Lösungen von \(\sigma(n)= \sigma(n+1)\), ist \(\sum \sigma_k(n)/n!\) irrational, Cullen’sche Zahlen, Teilbarkeitseigenschaften von Fakultät und Binomialkoeffizienten, Carmicheal’s Vermutung, und viele andere).

C) Additive Zahlentheorie (Scholz’sche Vermutung, für welche \(N\) ist \(N= \sum a_i x_i\) unlörbar?, Folgen \(\{a_i\}\) für die alle \(a_i+a_j\) verschieden sind, perfekte Differenzmengen, Fragen über additive Basen, ‘maximal sum-free sets”, Damen auf dem Schachbrett, Darstellung durch vier teilerfremde Quadrate, u.a.).

D) Diophantische Gleichungen (im Umfeld der Fermat’schen Vermutung, spezielle diophantische Gleichungen, Catalan’s Vermutung, Darstellung durch Stammbrüche, und viele andere).

E) Folgen ganzer Zahlen (dünne Folgen \(\{a_i\}\), so daß \(p+a_i=n\) für große \(n\) lösbar wird, Konvergenz von \(\sum(-1)^n\cdot n/p_n\), Umfeld des van der Waerden’schen Satzes über arithmetische Progressionen, u.v.a.).

F) Vermischte Probleme.

Ein ausführliches Namens- und Sachverzeichnis beschließen das Buch. Der Verfasser hat dankenswerterweise eine umfangreiche Liste (20 pp.) von Errata und Addenda zu diesem Buch zusamemngestellt, die u.a. weitere Literaturverweise und weitere Ergebnisse numerischer Rechnungen enthalten.

### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

00A07 | Problem books |

11Axx | Elementary number theory |

11Dxx | Diophantine equations |

11A05 | Multiplicative structure; Euclidean algorithm; greatest common divisors |

11A41 | Primes |

11A25 | Arithmetic functions; related numbers; inversion formulas |

11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |

11N05 | Distribution of primes |

11B13 | Additive bases, including sumsets |

11B25 | Arithmetic progressions |

11D41 | Higher degree equations; Fermat’s equation |

11D85 | Representation problems |

05A05 | Permutations, words, matrices |

### Keywords:

unsolved problems; collection of problems; prime number theory; divisibility; additive number theory; diophantine equations; sequences of integers; bibliography### Online Encyclopedia of Integer Sequences:

Fermat numbers: a(n) = 2^(2^n) + 1.No-3-in-line problem: number of inequivalent ways of placing 2n points on an n X n grid so that no 3 are in a line.

Lucky numbers.

a(n) = solution to the postage stamp problem with 3 denominations and n stamps.

a(n) is the solution to the postage stamp problem with 4 denominations and n stamps.

a(n) is the solution to the postage stamp problem with 5 denominations and n stamps.

a(n) is the solution to the postage stamp problem with 6 denominations and n stamps.

a(n) = solution to the postage stamp problem with n denominations and 2 stamps.

a(n) is the solution to the postage stamp problem with n denominations and 3 stamps.

a(n) is the solution to the postage stamp problem with n denominations and 4 stamps.

a(n) is the solution to the postage stamp problem with n denominations and 5 stamps.

a(n) = solution to the postage stamp problem with n denominations and 6 stamps.

Triangular numbers of form a(a+1)(a+2).

Wieferich primes: primes p such that p^2 divides 2^(p-1) - 1.

Numbers n such that phi(sigma(n)) = n.

Taxi-cab numbers: sums of 2 cubes in more than 1 way.

Numbers k such that k! - (k-1)! + (k-2)! - (k-3)! + ... - (-1)^k*1! is prime.

Smallest number that takes n steps to reach 1 under iteration of sum-of-squares-of-digits map (= smallest ”happy number” of height n).

Numbers k such that phi(k) = phi(k+1).

Image of n under the map n->n/2 if n even, n->3n-1 if n odd.

Numbers k such that phi(k) = phi(k+2).

A self-generating sequence: every positive integer occurs as a(i)-a(j) for a unique pair i,j.

a(1)=2, a(2)=3; for n >= 3, a(n) is smallest number that is uniquely of the form a(j) + a(k) with 1 <= j < k < n.

Davenport-Schinzel numbers of degree 4 on n symbols.

Segmented numbers, or prime numbers of measurement.

Prime numbers of measurement.

Cullen numbers: a(n) = n*2^n + 1.

Least number k such that phi(k) = m, where m runs through the values (A002202) taken by phi.

a(n) = floor(3^n / 2^n).

The game of Mousetrap with n cards (given n letters and n envelopes, how many ways are there to fill the envelopes so that at least one letter goes into its right envelope?).

The game of Mousetrap with n cards: the number of permutations of n cards in which 2 is the only hit.

Markoff (or Markov) numbers: union of positive integers x, y, z satisfying x^2 + y^2 + z^2 = 3*x*y*z.

Unitary perfect numbers: numbers k such that usigma(k) - k = k.

Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms.

a(1) = 1, a(2) = 3; for n >= 3, a(n) is smallest number that is uniquely of the form a(j) + a(k) with 1 <= j < k < n.

Egyptian fractions: number of solutions of 1 = 1/x_1 + ... + 1/x_n where 0 < x_1 <= ... <= x_n.

Egyptian fractions: number of solutions of 1 = 1/x_1 + ... + 1/x_n in positive integers.

Primitive weird numbers: weird numbers with no proper weird divisors.

Numbers k such that k! + 1 is prime.

Numbers n such that n! - 1 is prime.

Carmichael numbers: composite numbers n such that a^(n-1) == 1 (mod n) for every a coprime to n.

”Length” of aliquot sequence for n.

For n > 4, a(n) is the least integer > a(n-1) with precisely two representations a(n) = a(i) + a(j), 1 <= i < j < n; and a(n) = n for n=1..4.

a(n) (n>6) is least integer > a(n-1) with precisely three representations a(n) = a(i) + a(j), 1 <= i < j < n, a(n) = n for n=1..6.

One of the basic cycles in the x->3x-1 (x odd) or x/2 (x even) problem.

One of the basic cycles in the x->3x-1 (x odd) or x/2 (x even) problem.

n! is a nontrivial product of factorials. It is conjectured that the list is complete.

Congruent numbers: positive integers k for which there exists a right triangle having area k and rational sides.

Values of phi(n) = phi(n+1).

Szekeres’s sequence: a(n)-1 in ternary = n-1 in binary; also: a(1) = 1, a(2) = 2, and thereafter a(n) is smallest number k which avoids any 3-term arithmetic progression in a(1), a(2), ..., a(n-1), k.

Sociable numbers: smallest member of each cycle (conjectured).

Left factorials: !n = Sum_{k=0..n-1} k!.

The smaller of a betrothed pair.

The larger of a betrothed pair.

Numbers j such that the average of the divisors of j is an integer: sigma_0(j) divides sigma_1(j). Alternatively, numbers j such that tau(j) (A000005(j)) divides sigma(j) (A000203(j)).

Numbers that are the sum of two 4th powers in more than one way (primitive solutions).

Numbers that are the sum of two positive cubes in at least three ways (primitive solutions).

Numbers that are the sum of two cubes in at least four ways (primitive solutions).

’Core’ alternating sign n X n matrices, i.e., those that are not ’blown up’ from a smaller matrix by inserting row i, column j with a_ij = 1 and all other entries in that row and column equal to 0.

Numbers k such that k^4 is a primitive sum of 3 positive fourth powers.

Hofstadter-Conway \(10000 sequence: a(n) = a(a(n-1)) + a(n-a(n-1)) with a(1) = a(2) = 1\)

Davenport-Schinzel numbers of degree n on 3 symbols.

Davenport-Schinzel numbers of degree n on 4 symbols.

Davenport-Schinzel numbers of degree n on 5 symbols.

Minimal span of set of n elements with no 3-term arithmetic progression.

Class 1+ primes: primes of the form 2^i*3^j - 1 with i, j >= 0.

Class 2+ primes (for definition see A005105).

Class 3+ primes (for definition see A005105).

Class 4+ primes (for definition see A005105).

Class 2- primes (for definition see A005109).

Class 3- primes (for definition see A005109).

Class 4- primes (for definition see A005109).

Smallest prime in class n (sometimes written n+) according to the Erdős-Selfridge classification of primes.

Let i, i+d, i+2d, ..., i+(n-1)d be an n-term arithmetic progression of primes; choose the one which minimizes the last term; then a(n) = last term i+(n-1)d.

Self-contained numbers: odd numbers k whose Collatz sequence contains a higher multiple of k.

Hofstadter Q-sequence: a(1) = a(2) = 1; a(n) = a(n-a(n-1)) + a(n-a(n-2)) for n > 2.

Odd abundant numbers (odd numbers m whose sum of divisors exceeds 2m).

Primorial plus 1 primes: primes p such that 1 + product of primes up to p is prime.

Barriers for omega(n): numbers n such that, for all m < n, m + omega(m) <= n.

Numbers k such that k and k+1 have the same number of divisors.

Numbers n such that n, n+1 and n+2 have the same number of divisors.

Irregular triangle of Section I numbers. Row n contains numbers k with 2^n < k < 2^(n+1) and phi^n(k) = 2, where phi^n means n iterations of Euler’s totient function.

P-positions in Epstein’s Put or Take a Square game.

N-positions in Epstein’s Put or Take a Square game.

A self-generating sequence.

A self-generating sequence: start with 1 and 2, take all sums of any number of successive previous elements and adjoin them to the sequence. Repeat!

A self-generating sequence: start with 2 and 3, take all products of any 2 previous elements, subtract 1 and adjoin them to the sequence.

The (Mahler-Popken) complexity of n: minimal number of 1’s required to build n using + and *.

Record gaps between primes.

Betrothed (or quasi-amicable) numbers.

Nontotients: even numbers k such that phi(m) = k has no solution.

Numbers having divisors d,e with d < e < 2*d.

Davenport-Schinzel numbers of degree 5 on n symbols.

Davenport-Schinzel numbers of degree 6 on n symbols.

Mian-Chowla sequence (a B_2 sequence): a(1) = 1; for n>1, a(n) = smallest number > a(n-1) such that the pairwise sums of elements are all distinct.

Conway-Guy sequence: a(n + 1) = 2a(n) - a(n - floor( 1/2 + sqrt(2n) )).

a(n) = solution to the postage stamp problem with n denominations and 7 stamps.

a(n) = solution to the postage stamp problem with n denominations and 8 stamps.

a(n) = solution to the postage stamp problem with n denominations and 9 stamps.

Starts 0, 4 and contains no 3-term arithmetic progression.

Maximal number of edges in a b^{hat} graceful graph with n nodes.

Smallest number of complexity n: smallest number requiring n 1’s to build using + and *.

3-perfect (triply perfect, tri-perfect, triperfect or sous-double) numbers: numbers such that the sum of the divisors of n is 3n.

Lexicographically earliest increasing nonnegative sequence that contains no 4-term arithmetic progression.

Pseudoprimes to base 3.

Pseudoprimes to base 5.

Pseudoprimes to base 6.

Pseudoprimes to base 7.

Pseudoprimes to base 10.

Weird numbers: abundant (A005101) but not pseudoperfect (A005835).

The ”amusical permutation” of the nonnegative numbers: a(2n)=3n, a(4n+1)=3n+1, a(4n-1)=3n-1.

a(n) = 2*n/3 for n divisible by 3, otherwise a(n) = round(4*n/3). Or, equivalently, a(3*n-2) = 4*n-3, a(3*n-1) = 4*n-1, a(3*n) = 2*n.

The Collatz or 3x+1 map: a(n) = n/2 if n is even, 3n + 1 if n is odd.

Image of n after 3k iterates of ’3x+1’ map (k large).

Limit of the image of n after 2k iterates of ‘(3x+1)/2’ map as k grows.

a(n) = 2^(n-1)*(2^n - 1), n >= 0.

Numbers k such that k and k+1 are prime powers.

Start of first run of n consecutive integers with same number of divisors.

Number of halving and tripling steps to reach 1 in ’3x+1’ problem, or -1 if 1 is never reached.

Numbers k such that k, k+1, k+2 and k+3 have the same number of divisors.

Number of halving steps to reach 1 in ’3x+1’ problem, or -1 if this never happens.

Number of tripling steps to reach 1 from n in ’3x+1’ problem, or -1 if 1 is never reached.

Primorial -1 primes: primes p such that -1 + product of primes up to p is prime.

a(1)=4, a(2)=5; thereafter a(n) is smallest number that is greater than a(n-1) and having a unique representation as a(j) + a(k) for j<k.

Step at which n is expelled in Kimberling’s puzzle (A035486).

Numbers k such that phi(k) = phi(sigma(k)).

Even pseudoprimes (or primes) to base 2: even n that divide 2^n - 2.

Euler pseudoprimes: composite numbers n such that 2^((n-1)/2) == +-1 (mod n).

Composite numbers k such that k == +-1 (mod 8) and 2^((k-1)/2) == 1 (mod k).

Primitive congruent numbers.

a(n) = smallest k such that phi(n+k) = phi(k).

Leech’s tree-labeling problem for n nodes.

a(1)=2, a(2)=5; for n >= 3, a(n) is smallest number which is uniquely of the form a(j) + a(k) with 1 <= j < k < n.

a(n) = Sum_{k=1..n} k!.

a(n) = first n-fold perfect (or n-multiperfect) number.

2-hyperperfect numbers: n = 2*(sigma(n) - n - 1) + 1.

Number of winning (or reformed) decks at Mousetrap.

Number of once reformable permutations of {1,2,...,n}.

Happy numbers: numbers whose trajectory under iteration of sum of squares of digits map (see A003132) includes 1.

Number of lattice points inside circle of radius n is 4(a(n)+n)-3.

Number of steps for aliquot sequence for n to converge to 0, or -1 if it never reaches 0.

Minimal difference s(n) between beginning and end of n consecutive large primes (n-tuplet) permitted by divisibility considerations.

Primitive parts of Pell numbers.

3x+1 sequence starting at 97.

3x+1 sequence starting at 63.

3x+1 sequence starting at 95.

3x+1 sequence starting at 81.

3x+1 sequence starting at 57.

3x+1 sequence starting at 39.

3x+1 sequence starting at 87.

3x + 1 sequence starting at 33.

3x+1 sequence starting at 99.

3x+1 sequence starting at 51.

3x+1 sequence starting at 27.

Aliquot sequence starting at 30.

Aliquot sequence starting at 42.

Aliquot sequence starting at 60.

Aliquot sequence starting at 138.

Aliquot sequence starting at 150.

Aliquot sequence starting at 168.

Aliquot sequence starting at 180.

Aliquot sequence starting at 276.

3x - 1 sequence starting at 36.

x->x/2 if x even, x->3x-1 if x odd.

3x - 1 sequence starting at 66.

x->x/2 if x even, x->3x-1 if x odd.

Trajectory of 84 under the map x -> x/2 for x even, x -> 3x - 1 for x odd.

x -> x/2 if x even, x -> 3x - 1 if x odd.

x->x/2 if x even, x->3x-1 if x odd.

x->x/2 if x even, x->3x-1 if x odd.

x->x/2 if x even, x->3x-1 if x odd.

x->x/2 if x even, x->3x-1 if x odd.

(1 + number of halving and tripling steps to reach 1 in the Collatz (3x+1) problem), or -1 if 1 is never reached.

Taxicab, taxi-cab or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 positive integral cubes in n ways.

Levine’s sequence. First construct a triangle as follows. Row 1 is {1,1}; if row n is {r_1, ..., r_k} then row n+1 consists of {r_k 1’s, r_{k-1} 2’s, r_{k-2} 3’s, etc.}; sequence consists of the final elements in each row.

Number of numbers m with Euler phi(m) = n.

Aliquot sequence starting at 552.

Aliquot sequence starting at 564.

Aliquot sequence starting at 660.

Aliquot sequence starting at 966.

Aliquot sequence starting at 1074.

Aliquot sequence starting at 1134.

Nim-Grundy function for Take-a-Square (or Subtract-a-Square) game.

Nim function for Take-a-Factorial-Game (a subtraction game).

Nim function for Take-a-Fibonacci-Game (a subtraction game).

Nim function for Take-a-Prime (or Subtract-a-Prime) Game.

a(n) = 2^n mod n.

Numbers k such that k^2 contains exactly 2 distinct digits.

Numbers k such that k^2 contains exactly 2 different digits, excluding 10^m, 2*10^m, 3*10^m.

Numbers that are the sum of two 4th powers in more than one way.

Numbers that are the sum of two positive cubes in at least three ways (all solutions).

Squares using at most two distinct digits, not ending in 0.

Let Dedekind’s psi(m) = product of (p+1)p^(e-1) for primes p, where p^e is a factor of m. Iterating psi(m) eventually results in a number of form 2^a*3^b. a(n) is the smallest number that requires n steps to reach such a number.

Let Dedekind’s psi(m) = product of (p+1)p^(e-1) for primes p, where p^e is a factor of m. Iterating psi(m) eventually results in a number of form 2^a*3^b. a(n) is the number of steps to reach such a number.

Fermat primes: primes of the form 2^(2^k) + 1, for some k >= 0.

Nim-values for the impartial game Take-a-Triangle.

Numbers that are the sum of two positive cubes in at least four ways (all solutions).

Complexity of n: number of 1’s required to build n using +, * and ^.

Numbers k such that the set of prime divisors of k is equal to the set of prime divisors of sigma(k).

4-perfect (quadruply-perfect or sous-triple) numbers: sum of divisors of n is 4n.

Triangle of numbers of permutations eliminating just k cards out of n in game of Mousetrap.

Triangle read by rows of numbers of permutations eliminating just card k out of n in game of Mousetrap.

Number of terms in longest arithmetic progression of consecutive primes starting at n-th prime (conjectured to be unbounded).

0-additive sequence: not the sum of any previous pair.

Numbers that are in both the 1-additive and 0-additive sequences (A002858 and A033627).

Numbers that are not the sum of two distinct Ulam numbers.

Numbers k such that sigma(phi(k)) = sigma(k) where sigma is the sum of divisors function A000203 and phi is the Euler totient function A000010.

Deficiency of n, or 2n - (sum of divisors of n).

a(n) is the number of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1).

Maximal number of residue classes mod n such that no subset adds to 0.

Wolstenholme quotient W_p = (binomial(2p-1,p) - 1)/p^3 for prime p=A000040(n).

Number of ways to write n! as a product of smaller factorials each greater than 1.

Numbers k such that k! can be written as the product of smaller factorials.

Hyperperfect numbers: x such that x = 1 + k*(sigma(x)-x-1) for some k > 0.

n-th term of A034897 is an a(n)-hyperperfect number.

Kimberling’s expulsion array read by antidiagonals.

Happy primes: primes that eventually reach 1 under iteration of ”x -> sum of squares of digits of x”.

Lower of pair of consecutive happy numbers.

Upper of pair of consecutive happy numbers.

Numbers that eventually reach 1 under ”x -> sum of cubes of digits of x”.

Active part of Kimberling’s expulsion array as a triangular array.

Least inverse of A015910: smallest integer k > 0 such that 2^k mod k = n, or 0 if no such k exists.

Triangle of numbers arising from Gilbreath’s conjecture: successive absolute differences of primes (read by antidiagonals upwards, omitting the initial row of primes).

Array of numbers read by upward antidiagonals, arising from Gilbreath’s conjecture: leading row lists the primes; the following rows give absolute values of differences of previous row.

Position of first term > 2 in n-th row of Gilbreath array shown in A036262.

Number of ways of arranging row n of the prime pyramid.

From a subtractive Goldbach conjecture: odd primes that are not cluster primes.

From a subtractive Goldbach conjecture: cluster primes.

Largest subset of integers [ 1...n ] such that no member divides two others.

Odd values of n > 1 for which there are n-hyperperfect numbers.

Numbers n > 2 such that n - 2^k is a prime for all k > 0 with 2^k < n.

Schur’s numbers (version 2).

Numbers k such that sigma(k) == 2 (mod k).

Numbers n such that 6n+1, 12n+1 and 18n+1 are all primes.

Numbers n for which floor((3/2)^n) is composite.

Numbers (not of the form 9n+-4) that were conjectured not to be the sum of 3 positive or negative cubes. However, further research suggests that this sequence is in fact empty.

Fortunate primes (A005235) in numerical order with duplicates removed.

Least n-digit ’happy’ prime.

Numbers k such that 4*5^k - 1 is prime.

Numbers k such that 6*7^k - 1 is prime.

Good primes (version 1): prime(n)^2 > prime(n-1)*prime(n+1).

x such that y^2 = C(x,0) + C(x,1) + C(x,2) + C(x,3) is soluble.

y such that y^2 = C(x,0) + C(x,1) + C(x,2) + C(x,3) is soluble.

Smallest positive number that can be written in n ways as a sum of two (not necessarily positive) cubes.

Smallest positive number that can be written in n ways as a sum of two (not necessarily positive) coprime cubes.

Euler-Jacobi pseudoprimes: 2^((n-1)/2) == (2 / n) mod n, where (2 / n) is a Jacobi symbol.

Numbers n such that 2^n - 1 is divisible by a square > 1.

a(n) = gcd(n - 1, phi(n)).

Numbers m such that 2*phi(m) = phi(m+1).

Smallest k such that phi(k+n)=2*phi(k).

Lexicographically earliest prime pyramid, read by rows.

Numbers k such that k*sigma(k) == 2 (mod phi(k)).

Numbers m for which there exist d(1),...,d(m), each in {0,1}, such that Sum_{i=1..m-k} d(i)*d(i+k) is odd for all k=0,...,m-1.

a(n) = solution to the postage stamp problem with 7 denominations and n stamps.

a(n) = solution to the postage stamp problem with 8 denominations and n stamps.

Numbers k such that 1 + 4 + 9 + ... + k^2 = 1 + 2 + 3 + ... + m for some m.

Number of ways of representing n as the sum of one or more consecutive primes.

Smallest positive integer that can be expressed as the sum of consecutive primes in exactly n ways.

Integers that can be expressed as the sum of consecutive primes in exactly 1 way.

Integers that can be expressed as the sum of consecutive primes in exactly 2 ways.

Integers that can be expressed as the sum of consecutive primes in exactly 3 ways.

Integers that can be expressed as the sum of consecutive primes in exactly 4 ways.

Integers that can be expressed as the sum of consecutive primes in exactly 5 ways.

Integers that can be expressed as the sum of consecutive primes in exactly 6 ways.

Boris Stechkin’s function.

a(n) = n/A055032(n).

Norms of Gaussian primes.

Number of Gaussian primes of successive norms (indexed by A055025).

Number of inequivalent Gaussian primes of successive norms (indexed by A055025).

Number of Gaussian primes of norm n.

Number of inequivalent Gaussian primes of norm n.

(Sum(m^(p-1),m=1..p-1)+1)/p as p runs through the primes.

Numerator of (Sum(m^(n-1),m=1..n-1)+1)/n.

Denominator of (Sum(m^(n-1),m=1..n-1)+1)/n.

Numbers expressible as the sum of no more than 5 squares of composite numbers, allowing repetitions.

Write n as a sum of terms of the form (p^2-1)/24 where p is a prime > 4; sequence gives those n which require at least 4 terms.

Least m such that phi(m) = n!.

Norms of Eisenstein-Jacobi primes.

Number of Eisenstein-Jacobi primes of successive norms (indexed by A055664).

Number of inequivalent Eisenstein-Jacobi primes of successive norms (indexed by A055664).

Number of Eisenstein-Jacobi primes of norm n.

Number of inequivalent Eisenstein-Jacobi primes of norm n.

a(n) = n * a(n-1) - Sum_{k=1..n-2} a(k) with a(1) = 0 and a(2) = 1.

Initial prime in first sequence of n primes congruent to 3 modulo 4.

Initial prime in first sequence of n consecutive primes congruent to 1 modulo 6.

Initial prime in first sequence of n consecutive primes congruent to 5 modulo 6.

Initial prime in first sequence of n primes congruent to 1 modulo 4.

Freestyle perfect numbers n = Product_{i=1,..,k} f_i^e_i where 1 < f_1 < ... < f_k, e_i > 0, such that 2n = Product_{i=1,..,k} (f_i^(e_i+1)-1)/(f_i-1).

Numbers n such that the binomial coefficient C(2n,n) is not divisible by the square of an odd prime.

Smallest term in any sequence of n consecutive Smith numbers (A006753).

Numbers k such that k and k+1 are powerful numbers.

Numbers that are not congruent to 4 or 5 mod 9.

Value of y of the solution to x^3 + y^3 + z^3 = A060464(n) (numbers not 4 or 5 mod 9) with smallest |z| and smallest |y|, 0 <= |x| <= |y| <= |z|.

Value of z of the solution to x^3 + y^3 + z^3 = A060464(n) (numbers not 4 or 5 mod 9) with smallest |z| and smallest |y|, 0 <= |x| <= |y| <= |z|.

Odd powerful numbers.

Erdos primes: primes p such that all p-k! for 1<=k!<p are composite.

Floor(4^n / 3^n).

Sum of n/p^k over all maximal prime-power divisors of n.

Conjectured list of positive numbers which are not of the form r^i-s^j, where r,s,i,j are integers with i>1, j>1.

Numbers n such that sigma(n) divides sigma(phi(n)).

Start with a(0)=1, a(1)=4, a(2)=3, a(3)=2; for n>=3, a(n+1) = max_i (a(i)+a(n-i)).

Start with a(0)=1, a(1)=4, a(2)=3, a(3)=2; for n>=3, a(n+1) = mex_i (a(i)+a(n-i)), where mex means smallest nonnegative missing number.

Start with a(0)=1, a(1)=4, a(2)=3, a(3)=2; for n>=3, a(n+1) = mex_i (nim-sum a(i)+a(n-i)), where mex means smallest nonnegative missing number.

Primes of the form floor((3/2)^k).

a(n) = maximum length of a subset in {1,..,n} whose integers have pairwise LCM not exceeding n.

k values for self-contained numbers.

n* values for self-contained numbers.

Let S = a subset of {n,n+1,n+2,..,n+m}, number of parts of S is more than 1. a(n) = smallest number m such that the product of parts of S is a square.

p and q are primes. p^x - q^y = 2^h. ( p != q, h=>0, x>1, y>1 ) : sequence gives p^x arranged in increasing order.

Barriers for bigomega(n): numbers n such that, for all m < n, m + bigomega(m) <= n.

Values of floor((3/2)^n) that are composite.

n for which floor((3/2)^n) is prime.

Values of floor((4/3)^n) that are composite.

n for which floor((4/3)^n) is prime.

Smallest positive integer m where 1^n+2^n+3^n+...+m^n is greater than or equal to (m+1)^n.

Least k such that floor((1+1/k)^n) is prime.

Full list of counterexamples for the k=3 version of the malicious apprentice problem.

Aliquot sequence starting at 1521.

Aliquot sequence starting at 570.

Conjectured list of positive numbers which are not of the form r^i - s^j, where r,s,i,j are integers with r>0, s>0, i>1, j>1.

a(n) = solution to the postage stamp problem with n denominations and 10 stamps.

Numbers n such that n! is a product of distinct factorials k!*l!*m!*... with k, l, m, etc. < n.

Slowest-growing sequence of primes whose reciprocals sum to 1.

Reduced Collatz function R applied to the odd integers: a(n) = R(2n-1), where R(k) = (3k+1)/2^r, with r as large as possible.

Odd numbers that cannot be expressed as 2^k - 3^m where k and m are integers.

Number of solutions to Pillai’s equation a^x - b^y = n, with a>0, b>0, x>1, y>1.

Numbers n which appear to have a unique representation as the difference of two perfect powers; that is, there is only one solution to Pillai’s equation a^x - b^y = n, with a>0, b>0, x>1, y>1.

Numbers n which appear to have a unique representation as the difference of two perfect powers and those powers are both 2; that is, there is only one solution to Pillai’s equation a^x - b^y = n, with a>0, b>0, x>1, y>1 and that solution has x=y=2.

n which appear to have a unique representation as the difference of two perfect powers and one of those powers is odd; that is, there is only one solution to Pillai’s equation a^x - b^y = n, with a>0, b>0, x>1, y>1 and that solution has odd x or odd y (or both odd).

Smallest powerful number (definition 1) such that a(n)+n is also powerful.

The smaller of a pair of powerful numbers (A001694) that differ by 2.

Differences of consecutive powerful numbers (definition 1).

Numerator of Product_{i=1..n} (p_i+1)/(p_i-1) where p_i is the i-th prime.

Denominator of Product_{i=1..n} (p_i+1)/(p_i-1). Numerators are in A078559.

Class 5- primes (for definition see A005109).

Class 6- primes (for definition see A005109).

Class 7- primes.

Class 8- primes.

Class 9- primes.

Class 10- primes.

Class 11- primes.

Class 5+ primes (for definition see A005105).

Class 6+ primes.

Class 7+ primes.

Class 8+ primes.

Class 9+ primes.

Class 10+ primes.

Class 11+ primes.

a(n) = n-th prime of class 12- according to the Erdős-Selfridge classification.

a(n) = n-th prime of class 13- according to the Erdős-Selfridge classification.

Erroneous version of A005047.

Class 12+ primes.

Nondecreasing gaps between primes.

Indices of primes where nondecreasing gaps occur.

Let p = n-th prime; a(n) = number of pairs (i,j) with 0 < i < p, 0 < j < p such that ij == 1 mod p and i and j have opposite parity.

a(n) is the smallest m such that m > A005235(n) and A002110(n)+m is prime.

Solutions n of max(m+d(m))=n+2 for m<n; d(m) is the number of divisors of m.

Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4).

Class 13+ primes.

Harmonic numbers (A001599) which are not perfect (A000396).

Numbers n such that n - 2^k is squarefree for all 1 <= 2^k < n.

a(n) = least k such that {n+1, n+2, n+3, ... n+k} has a subset the product of whose members with n is a square.

a(n) = least k such that {n+0, n+1, n+2, n+3, ... n+k} has a nonempty subset the product of whose members is a square.

Erroneous version of A005115.

Triangle read by rows in which row n gives the n-set obtained as the differences {b(n)-b(n-i), 0 <= i <= n-1}, where b() = A005318().

a(n) = least denominator Y of the proper fractions X/Y which need n or more terms as an Egyptian fraction.

a(n) = least numerator X of the proper fractions X/A097048(n) which need n or more terms as an Egyptian fraction.

Matrix T(m,x(1)), m>=1, x(1)>=2, read by antidiagonals, where T(m,x(1)) gives the position of the first noninteger term in the sequence defined by x(n)=(x(n-1)*(x(n-1)^m+n-1))/n for n>=2 with exponent m and the given starting value x(1).

Length of aliquot sequence for n, or -1 if aliquot sequence never cycles.

Length of transient part of aliquot sequence for n, or -1 if transient part is infinite.

Prime(n) such that 4 does not divide the difference between prime(n) and prime(n+1).

Numbers n such that (!n)/2 is prime, where !n = Sum_{k=0..n-1} k!.

a(n) = n-th prime of Erdős-Selfridge classification n-.

a(n) = n-th prime of Erdős-Selfridge classification n+.

Primes p such that p! - 1 is also prime.

Primes of the form p! + 1 where p is prime.

Positive integers which have a ”compact” representation using fewer decimal digits than just writing the number normally.

Conjectured greatest number k such that C(2k,k) is not divisible by any odd prime to the n-th power.

Number of lucky numbers <= n.

Changes in United States postal rates per ounce since 1863.

a(n) = (2^(semiprime(n)-1)) modulo (semiprime(n)^2).

Leading terms in rows obtained by repeatedly computing consecutive absolute differences, starting with the squares of prime numbers.

Near-multiperfects: numbers m such that abs(sigma(m) mod m) <= log(m).

Near-multiperfects with primes excluded, abs(sigma(m) mod m) <= log(m).

Near-multiperfects with primes and powers of 2 excluded, abs(sigma(m) mod m) <= log(m).

Near-multiperfects with primes, powers of 2 and 6 * prime excluded, abs(sigma(n) mod n) <= log(n).

Near-multiperfects with primes, powers of 2, 6 * prime and 2^n * prime excluded, abs(sigma(n) mod n) <= log(n).

Least number with complexity height of n, under integer complexity A005245.

Number of 1’s required to build n using {+,T} and parentheses, where T(i) = i*(i+1)/2.

Greedy odd Egyptian fraction representation of 1 (without repeats).

Triangle in which row n is the lexicographically earliest solution to the prime circle problem for 2n.

Least number k < n and coprime to n such that the largest term of the continued fraction of k/n is as small as possible.

Number of odd numbers k such that phi(k) = n, where n runs through the values (A002202) taken by phi.

Numbers m such that the equation phi(x) = m has even but no odd solutions.

100*a(n)+13 and 100*a(n)+27 are consecutive primes, i.e., a prime gap 14.

Numbers k that are in the range of both Euler’s phi function and the sigma function.

Number of admissible bases in the postage stamp problem for n denominations and h = 2 stamps.

Number of admissible basis in the postage stamp problem for n denominations and h = 3 stamps.

Number of admissible basis in the postage stamp problem for n denominations and h = 4 stamps.

Number of admissible basis in the postage stamp problem for n denominations and h = 5 stamps.

Number of admissible basis in the postage stamp problem for n denominations and h = 6 stamps.

Number of admissible basis in the postage stamp problem for n denominations and h = 7 stamps.

Numbers k such that phi(k+1) = 4*phi(k).

Primes p such that (p, p+2, p+6, p+12) is a prime quadruple.

Primes p such that (p, p+2, p+6, p+12, p+14, p+20) is a prime sextuple.

Numbers n such that phi(phi(n)) + sigma(sigma(n)) is a 4th power.

Numbers n such that phi(phi(n)) + sigma(sigma(n)) is an 8th power.

Length of shortest computation yielding n using addition, subtraction and multiplication (starting from 1).

Spoof-perfect numbers: Freestyle perfect numbers (A058007) which are not perfect numbers (A000396).

Prime numbers n such that 2n-1 and 3n-2 are prime.

Maximum value on Lower Shuffle Part of Kimberling’s Expulsion Array (A035486).

Numerators of positive rationals < 1 arranged by increasing sum of numerator and denominator then by increasing numerator.

Denominators of positive rationals < 1 arranged by increasing sum of numerator and denominator then by increasing numerator.

Stanley Sequence S(0,3).

Primes p such that A005245(p) < A005245(p-1)+1.

Number of ways (up to rotations and reflections) of arranging numbers 1 through 2n around a circle such that the sum of each pair of adjacent numbers is composite.

a(n) = (Sum(m^(n-1), m=1..n-1) + 1) modulo n.

John Leech’s example of a set of eleven distinct odd numbers the sum of whose reciprocals is 1.

The first of the five known sets of nine distinct odd numbers the sum of whose reciprocals is 1.

The second of the five known sets of nine distinct odd numbers the sum of whose reciprocals is 1.

The third of the five known sets of nine distinct odd numbers the sum of whose reciprocals is 1.

The fourth of the five known sets of nine distinct odd numbers the sum of whose reciprocals is 1.

The fifth of the five known sets of nine distinct odd numbers the sum of whose reciprocals is 1.

Allan Johnson’s set of 48 squarefree numbers whose reciprocals add to 1, with the property that each number has exactly two distinct prime factors.

a(n) = Sum_{m=1..n-1} m^(n-1) modulo n.

Numbers not representable as the sum of three heptagonal numbers.

Numbers not representable as the sum of three octagonal numbers.

Numbers not representable as the sum of three 9-gonal numbers.

Numbers not representable as the sum of three 10-gonal numbers.

Numbers not representable as the sum of three 11-gonal numbers.

Numbers not representable as the sum of three 12-gonal numbers.

Even multiply-perfect numbers divided by 2.

Integer values of sigma(n)/n that are prime.

Number of changes of parity in the Collatz trajectory of n.

a(n) = Fibonacci(n-1) * Fibonacci(n+1) * Fibonacci(n+2).

a(n) = Fibonacci(n)^3 + (-1)^n*Fibonacci(n-2).

a(n) = Fibonacci(n-1) * Fibonacci(n) * Fibonacci(n+2).

a(n) = Fibonacci(n)^3 + (-1)^n*Fibonacci(n+2).

Conjectured total number of times that k-n appears in the Collatz (3x+1) sequence of k for k = 1, 2, 3,....

Odd numbers n with the property that Collatz (3x+1) trajectory of n contains exactly four terms that are divisible by 5.

Primes p == 2 (mod 5) for which 4*p+1 is also prime.

Number of integers k < n whose Collatz (3x+1) trajectory contains n.

Antiharmonic numbers that are not squares.

Numbers c > 0 for which there exist integers a > 1 and b > 1 such that the equation a^x - b^y = c has two solutions in positive integers x, y.

Primitive, symmetric octuples of distinct numbers a,b,c,d,x,y,z,w with 0<a<b<c<d and a<x<y<z<w<d such that a^k + b^k + c^k + d^k = x^k + y^k + z^k + w^k, for k = 1,2,3.

Triangle read by rows: row n lists the smallest positive ideal multigrade of degree n, or 2n+2 zeros if none.

Primes p at which phi(p-1)/(p-1) reaches a new minimum, where phi is Euler’s totient function.

Number of 2’s in the expansion of 2^n in base 3.

Length k of the longest chain of primes p_1, p_2, ..., p_k such that p_1 is the n-th prime and p_{i+1} equals 2*p_i + 1 or 2*p_i - 1 for all i < k, the +/- sign depending on i.

Babbage quotients b_p = (binomial(2p-1, p-1) - 1)/p^2 with p = prime(n).

Aliquot sequence starting at 702.

Areas of triples of primitive Pythagorean triangles having the same area.

Least prime beginning a string, of length at least n, of consecutive primes which alternate between types 4*k+1 and 4*k+3 or 4*k+3 and 4*k+1.

Least prime beginning a string, of length at least n, of consecutive primes which alternate between types 6*k+1 and 6*k+5 or 6*k+5 and 6*k+1.

Find the first (maximal) string, of length exactly n, of consecutive primes that alternate between types 6*k+1 and 6*k+5 or 6*k+5 and 6*k+1. The first element is a(n).

Self referencing version of the ”Kimberling shuffle” sequence (see Comments).

Quasi-amicable pairs.

Largest positive x such that (x,x+n) is the smallest pair of quadratic residues with difference n, modulo any prime.

Numbers m such that m^2 + 1 has at most 2 prime factors.

Positive integers that cannot be expressed as the sum of at most 5 pairwise coprime squares.

Nonzero multiplicative persistence in base 10: number of iterations of ”multiply nonzero digits in base 10” needed to reach a number < 10.

Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 2.

Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 1458.

Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 128.

Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 2*4^6.

Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 2*5^6.

Numbers that cannot be written as the sum of 5 or less squares of composite numbers, allowing repetitions (complement of A055075).

Numbers k such that the Diophantine equation x^3 + y^3 + z^3 = k has nontrivial primitive parametric solutions.

Numbers k such that the Diophantine equation x^3 + y^3 + 2*z^3 = k has nontrivial primitive parametric solutions.

Smallest prime which is at the end of an arithmetic progression of exactly n primes.

Initial terms associated with the arithmetic progressions of primes of A354376.

Positive integers with binary expansion (b(1), ..., b(m)) such that Sum_{i = 1..m-k} b(i)*b(i+k) is odd for all k = 0..m-1.