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%I #38 Oct 31 2022 14:00:16
%S 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,
%T 20,40,4,64,4,52,16,16,16,100,4,16,16,80,4,64,4,40,40,16,4,136,10,40,
%U 16,40,4,80,16,80,16,16,4,160,4,16,40,74,16,64,4,40,16
%N Multiplicative with a(p^e) = 2e^2 + 2.
%C If n is odd, then a(n) is the number of non-isomorphic generalized quaternion rings over Z_n.
%H Charles R Greathouse IV, <a href="/A286779/b286779.txt">Table of n, a(n) for n = 1..10000</a>
%H Jose María Grau, C. Miguel and A. M. Oller-Marcen, <a href="https://arxiv.org/abs/1706.04760">Generalized Quaternion Rings over Z/nZ for an odd n</a>, arXiv:1706.04760 [math.RA], 2017.
%F Dirichlet g.f.: zeta(s)^4 / zeta(4*s). - _Werner Schulte_, Oct 27 2022
%p a:= n-> mul(2*i[2]^2+2, i=ifactors(n)[2]):
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Jul 07 2017
%t fa[n_] := fa[n] = FactorInteger[n]; phi[1] = 1; phi[p_, s_] := 2s^2 + 2;
%t phi[n_] := Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i, Length[fa[n]]}];
%t Table[phi[n], {n, 1, 44}]
%o (PARI) a(n)=my(f=factor(n)[,2]); prod(i=1,#f, 2*f[i]^2+2) \\ _Charles R Greathouse IV_, Jul 07 2017
%K nonn,mult,easy
%O 1,2
%A _José María Grau Ribas_, May 14 2017
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