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A086811
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Average (scaled by a certain explicit factor) over all integers k of a_k(n), the n-th coefficient of the k-th cyclotomic polynomial.
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0
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0, 3, 6, 16, 45, 126, 224, 1344, 684, 1116, 4752, 23760, 56784, 286944, 164664, 281472, 2449224, 7371648, 27086400, 160392960, 49635936, 68277888, 1049956992, 6077306880, 1252224000, 3240801792, 2083408128, 4066530048, 35225729280, 142745587200, 717382656000, 6279166033920, 2442775449600, 2080906813440, 2251759104000
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OFFSET
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1,2
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COMMENTS
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When n is odd the n-th term is an integer. If n is even then twice the n-th term is an integer. Conjecturally (Y. Gallot) the n-th term is always an integer. For n <= 128 this has been verified numerically by Yves Gallot. It is also an unproved conjecture due to H. Möller (1970) that no term of this sequence is negative.
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LINKS
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FORMULA
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Let M_k = k * Product_{prime p<=k} p. Let q be any prime > k. Then the k-th term (for k >= 2) is M_k * Sum_{d|M_k} ( a_d(k) + a_{d*q}(k) )/(2*d). The average of the k-th coefficient of the n-th cyclotomic polynomial is given by the k-th coefficient of this sequence divided by Zeta(2) * k * Product_{p<=k} (p+1). (Zeta(2) = Pi^2/6.) [See Section 8.3 in Moree and Hommerson (2003).]
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MAPLE
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with(numtheory):for k from 1 to 50 do; v := 1: w := 1:j := 1:z := 1:while ithprime(j)<=k do; v := v*ithprime(j); w := w*(1+1/ithprime(j)); z := z*(ithprime(j)+1); j := j+1; end do: v := v*k:z := z*k:q := ithprime(j):te := 0:for i from 1 to nops(divisors(v)) do; d := divisors(v)[i]; kl(x) := 1; for j from 1 to k do; if modp(d, j)=0 then kl(x) := taylor(kl(x)*(1-x^j)^mobius(d/j), x, k+1); end if; end do: te := te+coeff(kl(x), x, k)/d; kl(x) := 1; for j from 1 to k do; if modp(q*d, j)=0 then kl(x) := taylor(kl(x)*(1-x^j)^mobius(q*d/j), x, k+1); end if; end do: te := te+coeff(kl(x), x, k)/d; end do: zr := te/(2*w):print(k, zr*z):end do:
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CROSSREFS
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KEYWORD
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frac,nonn
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AUTHOR
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Pieter Moree (moree(AT)mpim-bonn.mpg.de), Aug 05 2003
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EXTENSIONS
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STATUS
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approved
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