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Approximate formulas for some functions of prime numbers. (English) Zbl 0122.05001


MSC:

11N05 Distribution of primes
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Online Encyclopedia of Integer Sequences:

Euler totient function phi(n): count numbers <= n and prime to n.
pi(n), the number of primes <= n. Sometimes called PrimePi(n) to distinguish it from the number 3.14159...
The composite numbers: numbers n of the form x*y for x > 1 and y > 1.
Smallest prime that begins with n.
Primorial numbers (second definition): n# = product of primes <= n.
Floor of the Chebyshev function theta(n): a(n) = floor(Sum_{primes p <= n } log(p)).
n such that pi(n) >= phi(n).
The (10^n)-th composite number.
Decimal expansion of -zeta’(2) (the first derivative of the zeta function at 2).
Decimal expansion of constant B3 (or B_3) related to the Mertens constant.
n-th prime appears n-th composite number times.
n-th composite number appears n-th prime times.
Decimal expansion of sum log(p)/p^2 over the primes p=2,3,5,7,11,...
Decimal expansion of constant C = maximum value that psi(n)/n reaches where psi(n)=log(lcm(1,2,...,n)) and lcm(1,2,...,n)=A003418(n).
Decimal expansion of constant C = maximum value that PrimePi(n)*log(n)/n reaches where PrimePi(n) is the number of primes less than or equal to n, A000720.
Decimal expansion of constant C = maximum value that sigma(n)*log(n^2)/n^2 reaches where sigma(n) = (sum of primes <= n), A034387.
Decimal expansion of -zeta’(3) (the first derivative of the zeta function at 3).
Decimal expansion of zeta’(-5) (the derivative of Riemann’s zeta function at -5) (negated).
Decimal expansion of -zeta’(4).
Decimal expansion of Rosser’s constant.
Define P = e^gamma*loglog(n) and Q = 3/loglog(n), where gamma is Euler’s constant A001620. Then a(n) = phi(n) - ceiling(n/(P + Q)), where phi(n) is Euler’s function A000010.
Decimal expansion of Sum_{p prime} log(p)/p^3.
Decimal expansion of Sum_{p prime} log(p)/p^4.
Decimal expansion of -Zeta’(2)/Zeta(2).
Integers n such that lcm(1, ..., n) > exp(n).
a(n) is the largest nonnegative integer m such that m >= pi(m)^(1 + 1/n).
a(n) is the largest nonnegative integer m such that m - pi(m) >= pi(m)^(1 + 1/n).