

A086811


Average (scaled by a certain explicit factor) over all integers k of a_k(n), the nth coefficient of the kth 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 nth term is an integer. If n is even then twice the nth term is an integer. Conjecturally (Y. Gallot) the nth 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 iten Koeffizienten der Kreisteilungspolynome, Math. Ann. 188 (1970), 2638.


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 kth term (for k>=2) is M_k*sum_{dM_k}(a_d(k)+a_{dq}(k))/(2d). The average of the kth coefficient of the nth cyclotomic polynomial is given by the kth 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)*(1x^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)*(1x^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)mpimbonn.mpg.de), Aug 05 2003


STATUS

approved



