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A086811 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. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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. Moller that no term of this sequence is negative.

REFERENCES

H. Moller, Über die i-ten Koeffizienten der Kreisteilungspolynome, Math. Ann. 188 (1970), 26-38.

LINKS

Table of n, a(n) for n=1..29.

Pieter Moree and Huib Hommersom, Value distribution of Ramanujan sums ...

FORMULA

Let M_k=k*prod_{p<=k}p, where p runs over the primes <=k. 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_{dq}(k))/(2d). 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 prod_{p<=k}(p+1) (Zeta(2)=pi^2/6).

MAPLE

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:

CROSSREFS

Sequence in context: A192676 A202846 A107269 * A106361 A113040 A244036

Adjacent sequences:  A086808 A086809 A086810 * A086812 A086813 A086814

KEYWORD

frac,nonn

AUTHOR

Pieter Moree (moree(AT)mpim-bonn.mpg.de), Aug 05 2003

STATUS

approved

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Last modified May 24 22:11 EDT 2017. Contains 287008 sequences.