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A064538
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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.
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5
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n) is a multiple of n+1. [From 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. [From Vladimir Shevelev, Dec 21 2011]
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REFERENCES
| V. S. Abramovich (Shevelev), Power sums of natural numbers, Kvant no.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, 1964 (and various reprintings), p. 804, Eq. 23.1.4.
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LINKS
| T. D. Noe, Table of n, a(n) for n = 0..1000
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].
Dr. Math, summing n^k.
E. S. Rowland, Sums of Consecutive Powers
Eric Weisstein's World of Mathematics, Power Sum
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EXAMPLE
| 1^3 + 2^3 + ... + x^3 = (x(x+1))^2/4 so a(3)=4.
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MAPLE
| A064538 := n -> denom((bernoulli(n+1, x)-bernoulli(n+1))/(n+1)):
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CROSSREFS
| Sequence in context: A164020 A057643 A073039 * A002790 A108951 A181822
Adjacent sequences: A064535 A064536 A064537 * A064539 A064540 A064541
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KEYWORD
| nonn,nice,easy
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AUTHOR
| Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 08 2001
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