A091398 - OEIS

%I #14 Oct 17 2022 07:06:07

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

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

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

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

%H Andrew Howroyd, <a href="/A091398/b091398.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) * log(phi)/(2*Pi^2) = 0.272559..., where phi is the golden ratio (A001622). - _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 23 2018

%Y Cf. A001622, A080891, A091379, A091396.

%K nonn,mult

%O 1,11

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