A035282 Expansion of zeta function of icosian ring (nonzero terms).

1, 5, 6, 10, 24, 21, 40, 30, 31, 60, 64, 50, 84, 120, 60, 50, 144, 120, 124, 85, 144, 200, 160, 126, 91, 180, 240, 240, 155, 204, 220, 300, 410, 320, 156, 264, 280, 210, 360, 300, 304, 384, 420, 170, 400, 504, 360, 300, 364, 384, 250, 400, 504, 960, 424, 720, 310

Offset: 1

Author

N. J. A. Sloane

Keywords

• easy
• nonn

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).

Nonzero terms of A078473. - Michel Marcus, Mar 03 2014

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

Crossrefs

Cf. A031363, A035187, A078472, A078473.

Programs

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}]]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 200], # > 0 &] (* Amiram Eldar, May 13 2022 *)