Kellner, Bernd C.; Sondow, Jonathan The denominators of power sums of arithmetic progressions. (English) Zbl 1423.11029 Integers 18, Paper A95, 17 p. (2018). The authors study the denominators of polynomials that represent the power sums of arithmetic progressions: \[ {\mathcal S}_{m,r}^n(x)=\sum_{k=0}^{x-1}(km+r)^n=r^n+(m+r)^n+\dots+((x-1)m+r)^n. \] They extend their earlier results on the case of power sum’s (when \(r=0,m=1\)). Specially, they give a simple explicit criterion for the integrality of the coefficients of these polynomials, and show further applications about the sequence of denominators of the Bernoulli polynomials. Reviewer: Péter Pál Pach (Budapest) Cited in 4 Documents MSC: 11B25 Arithmetic progressions 11B68 Bernoulli and Euler numbers and polynomials Keywords:Bernoulli polynomial; denominator Software:OEIS × Cite Format Result Cite Review PDF Full Text: arXiv Link Online Encyclopedia of Integer Sequences: a(n) is the smallest positive integer such that a(n)*(1^n + 2^n + ... + x^n) is a polynomial in x with integer coefficients. Positive integers k such that the derivative of the k-th Bernoulli polynomial B(k,x) contains only integer coefficients. Least number k such that all coefficients of k*B(n,x), the n-th Bernoulli polynomial, are integers. a(n) = denominator(Bernoulli_{n+1}(x) - Bernoulli_{n+1}). a(n) = denominator(Bernoulli_{n}(x)) / denominator(Bernoulli_{n}). a(n) = b(2*n-1)/b(2*n) where b(n) = A195441(n-1) = denominator(Bernoulli_{n}(x) - Bernoulli_{n}). a(n) = b(2*n)/b(2*n+1) where b(n) = denominator(Bernoulli_{n}(x)). Indices k such that A195441(k) = A195441(k+1). Numbers that appear in A195441 at least once for two consecutive indices. a(n) = (denominator of B(n,x)) / (the squarefree kernel of n+1), where B(n,x) is the n-th Bernoulli polynomial. Denominator of the second derivative of the n-th Bernoulli polynomial B(n,x). Positive integers k such that the second derivative of the k-th Bernoulli polynomial B(k,x) contains only integer coefficients. Positive integers k such that the third derivative of the k-th Bernoulli polynomial B(k, x) contains only integer coefficients. Positive integers k such that the fourth derivative of the k-th Bernoulli polynomial B(k, x) contains only integer coefficients. Positive integers k such that the fifth derivative of the k-th Bernoulli polynomial B(k, x) contains only integer coefficients. References: [1] G. Almkvist and A. Meurman, Values of Bernoulli polynomials and Hurwitz’s zeta function at rational points, C. R. Math. Acad. Sci. Soc. R. Can. 13 no. 2-3 (1991), 104-108. · Zbl 0731.11014 [2] A. Bazs´o and I. Mez˝o, On the coefficients of power sums of arithmetic progressions, J. Number Theory 153 (2015), 117-123. · Zbl 1365.11020 [3] A. Bazs´o, ´A. Pint´er, and H. M. Srivastava, A refinement of Faulhaber’s theorem concerning sums of powers of natural numbers, Appl. Math. Lett. 25 no. 3 (2012), 486-489. · Zbl 1285.11034 [4] H. Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM 240, Springer– Verlag, New York, 2007. · Zbl 1119.11002 [5] B. C. Kellner, On a product of certain primes, J. Number Theory 179 (2017), 126-141. · Zbl 1418.11045 [6] B. C. Kellner and J. Sondow, Power-sum denominators, Amer. Math. Monthly 124 (2017), 695-709. · Zbl 1391.11052 [7] N. E. Nørlund, Vorlesungen ¨uber Di↵erenzenrechnung, J. Springer, Berlin, 1924. · JFM 50.0318.04 [8] V. V. Prasolov, Polynomials, D. Leites, transl., 2nd edition, ACM 11, Springer-Verlag, Berlin, 2010. [9] A. M. Robert, A Course in p-adic Analysis, GTM 198, Springer-Verlag, New York, 2000. INTEGERS: 18 (2018)17 · Zbl 0947.11035 [10] H. G. Senge and E. G. Straus, PV-numbers and sets of multiplicity, Period. Math. Hungar. 3 (1973), 93-100. · Zbl 0248.12004 [11] N. J. A. Sloane, ed., The On-Line Encyclopedia of Integer Sequences, http://oeis.org. · Zbl 1044.11108 [12] C. L. Stewart, On the representation of an integer in two di↵erent bases, J. Reine Angew. Math. 319 (1980), 63-72. · Zbl 0426.10008 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.