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a(n) = Product_{ p | n } (1 + Legendre(-5,p) ).
2

%I #14 Oct 17 2022 07:05:45

%S 1,0,2,0,1,0,2,0,2,0,0,0,0,0,2,0,0,0,0,0,4,0,2,0,1,0,2,0,2,0,0,0,0,0,

%T 2,0,0,0,0,0,2,0,2,0,2,0,2,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,4,0,0,0,2,0,

%U 4,0,0,0,0,0,2,0,0,0,0,0,2,0,2,0,0,0,4,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,2,0,4

%N a(n) = Product_{ p | n } (1 + Legendre(-5,p) ).

%H Andrew Howroyd, <a href="/A091394/b091394.txt">Table of n, a(n) for n = 1..1000</a>

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5*sqrt(5)/(6*Pi) = 0.593135 . - _Amiram Eldar_, Oct 17 2022

%p with(numtheory); L := proc(n,N) local i,t1,t2; t1 := ifactors(n)[2]; t2 := mul((1+legendre(N,t1[i][1])),i=1..nops(t1)); end; [seq(L(n,-5),n=1..120)];

%t a[n_] := Times@@ (1+KroneckerSymbol[-5, #]& /@ FactorInteger[n][[All, 1]]);

%t Array[a, 105] (* _Jean-François Alcover_, Apr 08 2020 *)

%o (PARI) a(n)={my(f=factor(n)[, 1]); prod(i=1, #f, 1 + kronecker(-5, f[i]))} \\ _Andrew Howroyd_, Jul 25 2018

%Y Cf. A091379, A226162.

%K nonn,mult

%O 1,3

%A _N. J. A. Sloane_, Mar 02 2004