A078473 Expansion of zeta function of icosian ring.

1, 0, 0, 5, 6, 0, 0, 0, 10, 0, 24, 0, 0, 0, 0, 21, 0, 0, 40, 30, 0, 0, 0, 0, 31, 0, 0, 0, 60, 0, 64, 0, 0, 0, 0, 50, 0, 0, 0, 0, 84, 0, 0, 120, 60, 0, 0, 0, 50, 0, 0, 0, 0, 0, 144, 0, 0, 0, 120, 0, 124, 0, 0, 85, 0, 0, 0, 0, 0, 0, 144, 0, 0, 0, 0, 200, 0, 0, 160, 126, 91, 0, 0, 0, 0, 0, 0, 0

Offset

1,4

Comments

Let zetaI(s) be the zeta function of icosian ring: zetaI(s) = zetaQ(tau)(2s)*zetaQ(tau)(2s-1) where zetaQ(tau)(s) is defined in A035187. Then zetaI(s) = Sum_{n>=1} a(n)/n^(2s).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. 51 (1999), 1258-1276; arXiv preprint, arXiv:math/9904028 [math.MG], 1999.

Formula

Multiplicative with a(p^e) = (5^(e + 1) - 1)/4 if p = 5, (p^(e + 2) - 1)/(p^2 - 1) or 0 if p == 2 or 3 (mod 5) and e is even or odd, respectively, and Sum_{k=0..e} (k + 1)*(e - k + 1)*p^k if p == 1 or 4 (mod 5). - Amiram Eldar, May 13 2022

Mathematica

f[p_, e_] := Which[p == 5, (5^(e + 1) - 1)/4, (m = Mod[p, 5]) == 2 || m == 3, If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), 0], m == 1 || m == 4, Sum[(k + 1)*(e - k + 1)*p^k, {k, 0, e}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 13 2022 *)

Prog

(PARI) {a(n)=local(A); if(n<1, 0, A=direuler(p=2, n, 1/(1-X)/(1-kronecker(5, p)*X)); sumdiv(n, d, A[d]*d*A[n/d]))} /* Michael Somos, Jun 06 2005 */
(PARI) pf(p, r) = {if (p==5, (5^(r+1) -1)/4, if (((p % 5) == 2) || ((p % 5) == 3), if (!(r % 2), (p^(r+2) - 1)/(p^2-1), 0), if (((p % 5) == 1) || ((p % 5) == 4), sum(k=0, r, (k+1)*(r-k+1)*p^k))); ); }
a(n) = {my(f = factor(n)); prod(i=1, #f~, pf(f[i, 1], f[i, 2])); } \\ Michel Marcus, Mar 03 2014

Crossrefs

Cf. A035187, A035282 (nonzero terms of the sequence), A031363 (n for which a(n) is not zero), A078471.

Sequence in context: A102058 A231409 A031364 * A215833 A110800 A021645

Adjacent sequences: A078470 A078471 A078472 * A078474 A078475 A078476

Keyword

nonn,mult

Author

Benoit Cloitre, Dec 31 2002

Status

approved