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

Author

Benoit Cloitre, Dec 31 2002

Keywords

• nonn
• mult

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.

Crossrefs

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

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}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 13 2022 *)

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

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