A286779 Multiplicative with a(p^e) = 2e^2 + 2.

1, 4, 4, 10, 4, 16, 4, 20, 10, 16, 4, 40, 4, 16, 16, 34, 4, 40, 4, 40, 16, 16, 4, 80, 10, 16, 20, 40, 4, 64, 4, 52, 16, 16, 16, 100, 4, 16, 16, 80, 4, 64, 4, 40, 40, 16, 4, 136, 10, 40, 16, 40, 4, 80, 16, 80, 16, 16, 4, 160, 4, 16, 40, 74, 16, 64, 4, 40, 16

Offset: 1

Author

José María Grau Ribas, May 14 2017

Keywords

• nonn
• mult
• easy

Comments

If n is odd, then a(n) is the number of non-isomorphic generalized quaternion rings over Z_n.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Jose María Grau, C. Miguel and A. M. Oller-Marcen, Generalized Quaternion Rings over Z/nZ for an odd n, arXiv:1706.04760 [math.RA], 2017.

Programs

Maple

a:= n-> mul(2*i[2]^2+2, i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 07 2017

Mathematica

fa[n_] := fa[n] = FactorInteger[n]; phi[1] = 1; phi[p_, s_] := 2s^2 + 2;
phi[n_] := Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i, Length[fa[n]]}];
Table[phi[n], {n, 1, 44}]

PARI

a(n)=my(f=factor(n)[,2]); prod(i=1,#f, 2*f[i]^2+2) \\ Charles R Greathouse IV, Jul 07 2017

Formula

Dirichlet g.f.: zeta(s)^4 / zeta(4*s). - Werner Schulte, Oct 27 2022