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 A064538 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. 13
 1, 2, 6, 4, 30, 12, 42, 24, 90, 20, 66, 24, 2730, 420, 90, 48, 510, 180, 3990, 840, 6930, 660, 690, 720, 13650, 1092, 378, 56, 870, 60, 14322, 7392, 117810, 7140, 210, 72, 1919190, 103740, 8190, 1680, 94710, 13860, 99330, 9240, 217350, 9660, 9870, 10080, 324870 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n) is a multiple of n+1. - Vladimir Shevelev, Dec 20 2011 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 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 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 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 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 REFERENCES 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. LINKS Chai Wah Wu, Table of n, a(n) for n = 0..10000 (n = 0..1000 from T. D. Noe) V. S. Abramovich, Power sums of natural numbers, Kvant 5 (1973), 22-25. (in Russian) M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Bernd C. Kellner, On a product of certain primes, arXiv:1705.04303 [math.NT] 2017, J. Number Theory, 179 (2017), 126-141. Bernd C. Kellner, Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709. doi:10.4169/amer.math.monthly.124.8.695, arXiv:1705.03857 Bernd C. Kellner and Jonathan Sondow, The denominators of power sums of arithmetic progressions, arXiv:1705.05331 [math.NT], 2017, Integers, 18 (2018), article A95. Dr. Math, Summing n^k. R. Mestrovic, A congruence modulo n^3 involving two consecutive sums of powers and its applications, arXiv:1211.4570 [math.NT], 2012. R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4. G. Michon, Faulhaber's Formula on NUMERICANA.com. E. S. Rowland, Sums of Consecutive Powers V. Shevelev, A Short Proof of a Known Relation for Consecutive Power Sums, arXiv:0711.3692 [math.CA], 2007. Eric Weisstein's World of Mathematics, Power Sum Wikipedia, Faulhaber's Formula. FORMULA a(n) = (n+1)*A195441(n). - Jonathan Sondow, Nov 12 2015 A001221(a(n)/(n+1)) = A001222(a(n)/(n+1)). - Kieren MacMillan and Jonathan Sondow, Nov 20 2015 EXAMPLE 1^3 + 2^3 + ... + x^3 = (x(x+1))^2/4 so a(3)=4. 1^4 + 2^4 + ... + x^4 = x(x+1)(2x+1)(3x^2+3x-1)/30, therefore a(4)=30. MAPLE A064538 := n -> denom((bernoulli(n+1, x)-bernoulli(n+1))/(n+1)): # Peter Luschny, Aug 19 2011 # Formula of Kellner and Sondow (2017): a := proc(n) local s; s := (p, n) -> add(i, i=convert(n, base, p)); select(isprime, [\$2..(n+2)/(2+irem(n, 2))]); (n+1)*mul(i, i=select(p->s(p, n+1)>=p, %)) end: seq(a(n), n=0..48); # Peter Luschny, May 14 2017 MATHEMATICA a[n_] := Denominator[ Together[ (BernoulliB[n+1, x] - BernoulliB[n+1])/(n+1)] ]; Table[a[n], {n, 0, 44}] (* Jean-François Alcover, Feb 21 2012, after Maple *) PROG (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 (Sage) 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]) print [A064538(n) for n in (0..48)] # Peter Luschny, May 14 2017 (Python) from __future__ import division from sympy.ntheory.factor_ import digits, nextprime def A064538(n):     p, m = 2, n+1     while p <= (n+2)//(2+ (n% 2)):         if sum(d for d in digits(n+1, p)[1:]) >= p:             m *= p         p = nextprime(p)     return m # Chai Wah Wu, Mar 07 2018 CROSSREFS Cf. A195441, A256581, A286516, A286762, A286763. Sequence in context: A228099 A227955 A324922 * A002790 A108951 A181822 Adjacent sequences:  A064535 A064536 A064537 * A064539 A064540 A064541 KEYWORD nonn,nice,look,easy AUTHOR Floor van Lamoen, Oct 08 2001 STATUS approved

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Last modified October 15 23:54 EDT 2019. Contains 328038 sequences. (Running on oeis4.)