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%I #140 Oct 13 2023 10:02:41
%S 1,2,6,4,30,12,42,24,90,20,66,24,2730,420,90,48,510,180,3990,840,6930,
%T 660,690,720,13650,1092,378,56,870,60,14322,7392,117810,7140,210,72,
%U 1919190,103740,8190,1680,94710,13860,99330,9240,217350,9660,9870,10080,324870
%N 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.
%C a(n) is a multiple of n+1. - _Vladimir Shevelev_, Dec 20 2011
%C Let P_n(x) = 1^n + 2^n + ... + x^n = Sum_{i=1..n+1}c_i*x^i. Let P^*_n(x) = Sum_{i=1..n+1}(c_i/(i+1))*(x^(i+1)-x). Then b(n) = (n+1)*a(n+1)is the smallest positive integer such that b(n)*P^*_n(x) is a polynomial with integer coefficients. Proof follows from the recursion P_(n+1)(x) = x + (n+1)*P^*_n(x). As a corollary, note that, if p is the maximal prime divisor of a(n), then p<=n+1. - _Vladimir Shevelev_, Dec 21 2011
%C The recursion P_(n+1)(x) = x + (n+1)*P^*_n(x) is due to Abramovich (1973); see also Shevelev (2007). - _Jonathan Sondow_, Nov 16 2015
%C The sum S_m(n) = Sum_{k=0..n} k^m can be written as S_m(n) = n(n+1)(2n+1)P_m(n)/a(m) for even m>1, or S_m(n) = n^2*(n+1)^2*P_m(n)/a(m) for odd m>1, where a(m) is the LCM of the denominators of the coefficients of the polynomial P_m/a(m), i.e., the smallest integer such that P_m defined in this way has integer coefficients. (Cf. Michon link.) - _M. F. Hasler_, Mar 10 2013
%C a(n)/(n+1) is squarefree, by Faulhaber's formula and the von Staudt-Clausen theorem on the denominators of Bernoulli numbers. - _Kieren MacMillan_ and _Jonathan Sondow_, Nov 20 2015
%C a(n) equals n+1 times the product of the primes p <= (n+2)/(2+(n mod 2)) such that the sum of the base-p digits of n+1 is at least p. - _Bernd C. Kellner_ and _Jonathan Sondow_, May 24 2017
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprints), p. 804, Eq. 23.1.4.
%H Chai Wah Wu, <a href="/A064538/b064538.txt">Table of n, a(n) for n = 0..10000</a> (n = 0..1000 from T. D. Noe)
%H V. S. Abramovich, <a href="http://kvant.mccme.ru/1973/05/summy_odinakovyh_stepenej_natu.htm">Power sums of natural numbers</a>, Kvant 5 (1973), 22-25. (in Russian)
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Bernd C. Kellner, <a href="https://doi.org/10.1016/j.jnt.2017.03.020">On a product of certain primes</a>, J. Number Theory, 179 (2017), 126-141; arXiv:<a href="https://arxiv.org/abs/1705.04303">1705.04303</a> [math.NT], 2017.
%H Bernd C. Kellner, <a href="https://arxiv.org/abs/2310.01325">On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients</a>, 9 pp.; arXiv:2310.01325 [math.NT], 2023.
%H Bernd C. Kellner and Jonathan Sondow, <a href="https://doi.org/10.4169/amer.math.monthly.124.8.695">Power-Sum Denominators</a>, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:<a href="https://arxiv.org/abs/1705.03857">1705.03857</a> [math.NT], 2017.
%H Bernd C. Kellner and Jonathan Sondow, <a href="http://math.colgate.edu/~integers/s95/s95.pdf">The denominators of power sums of arithmetic progressions</a>, Integers 18 (2018), #A95, 17 pp.; arXiv:<a href="https://arxiv.org/abs/1705.05331">1705.05331</a> [math.NT], 2017.
%H Bernd C. Kellner and Jonathan Sondow, <a href="http://math.colgate.edu/~integers/v52/v52.pdf">On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits</a>, Integers 21 (2021), #A52, 21 pp.; arXiv:<a href="https://arxiv.org/abs/1902.10672">1902.10672</a> [math.NT], 2019.
%H Meng Fai Lim, <a href="https://arxiv.org/abs/2308.04099">On the p-divisibility of even K-groups of the ring of integers of a cyclotomic field</a>, arXiv:2308.04099 [math.NT], 2023.
%H Dr. Math, <a href="https://web.archive.org/web/20020219093559/http://mathforum.org/dr.math/problems/kijjaz11.24.98.html">Summing n^k</a>.
%H R. Mestrovic, <a href="http://arxiv.org/abs/1211.4570">A congruence modulo n^3 involving two consecutive sums of powers and its applications</a>, arXiv:1211.4570 [math.NT], 2012.
%H R. Mestrovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mestrovic/mes4.html">On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers</a>, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
%H G. Michon, <a href="http://www.numericana.com/answer/numbers.htm#faulhaber">Faulhaber's Formula</a> on NUMERICANA.com.
%H E. S. Rowland, <a href="https://ericrowland.github.io/investigations/sumsofpowers.html">Sums of Consecutive Powers</a>
%H Vladimir Shevelev, <a href="http://arxiv.org/abs/0711.3692">A Short Proof of a Known Relation for Consecutive Power Sums</a>, arXiv:0711.3692 [math.CA], 2007.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerSum.html">Power Sum</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Faulhaber's_formula">Faulhaber's Formula</a>.
%F a(n) = (n+1)*A195441(n). - _Jonathan Sondow_, Nov 12 2015
%F A001221(a(n)/(n+1)) = A001222(a(n)/(n+1)). - _Kieren MacMillan_ and _Jonathan Sondow_, Nov 20 2015
%F rad(a(n)) = A007947(a(n)) = A144845(n) = A324369(n+1) * A324370(n+1) * A324371(n+1). - _Bernd C. Kellner_, Oct 12 2023
%e 1^3 + 2^3 + ... + x^3 = (x(x+1))^2/4 so a(3)=4.
%e 1^4 + 2^4 + ... + x^4 = x(x+1)(2x+1)(3x^2+3x-1)/30, therefore a(4)=30.
%p A064538 := n -> denom((bernoulli(n+1,x)-bernoulli(n+1))/(n+1)): # _Peter Luschny_, Aug 19 2011
%p # Formula of Kellner and Sondow (2017):
%p a := proc(n) local s; s := (p,n) -> add(i,i=convert(n,base,p));
%p select(isprime,[$2..(n+2)/(2+irem(n,2))]);
%p (n+1)*mul(i,i=select(p->s(p,n+1)>=p,%)) end: seq(a(n), n=0..48); # _Peter Luschny_, May 14 2017
%t A064538[n_] := Denominator[ Together[ (BernoulliB[n+1, x] - BernoulliB[n+1])/(n+1)]];
%t Table[A064538[n], {n, 0, 44}] (* _Jean-François Alcover_, Feb 21 2012, after Maple *)
%o (PARI) a(n) = {my(vp = Vec(bernpol(n+1, x)-bernfrac(n+1))/(n+1)); lcm(vector(#vp, k, denominator(vp[k])));} \\ _Michel Marcus_, Feb 07 2016
%o (Sage)
%o A064538 = lambda n: (n+1)*mul([p for p in (2..(n+2)//(2+n%2)) if is_prime(p) and sum((n+1).digits(base=p)) >= p])
%o print([A064538(n) for n in (0..48)]) # _Peter Luschny_, May 14 2017
%o (Python)
%o from __future__ import division
%o from sympy.ntheory.factor_ import digits, nextprime
%o def A064538(n):
%o p, m = 2, n+1
%o while p <= (n+2)//(2+ (n% 2)):
%o if sum(d for d in digits(n+1,p)[1:]) >= p:
%o m *= p
%o p = nextprime(p)
%o return m # _Chai Wah Wu_, Mar 07 2018
%Y Cf. A144845, A195441, A256581, A286516, A286762, A286763, A324369, A324370, A324371.
%K nonn,nice,look,easy
%O 0,2
%A _Floor van Lamoen_, Oct 08 2001
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