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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)



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.


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


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

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


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).


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:


Sequence in context: A192676 A202846 A107269 * A106361 A113040 A244036

Adjacent sequences:  A086808 A086809 A086810 * A086812 A086813 A086814




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



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Last modified August 20 17:19 EDT 2017. Contains 290837 sequences.