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Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -25.
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%I #16 Nov 17 2023 17:51:20

%S 1,2,0,3,1,0,0,4,1,2,0,0,2,0,0,5,2,2,0,3,0,0,0,0,1,4,0,0,2,0,0,6,0,4,

%T 0,3,2,0,0,4,2,0,0,0,1,0,0,0,1,2,0,6,2,0,0,0,0,4,0,0,2,0,0,7,2,0,0,6,

%U 0,0,0,4,2,4,0,0,0,0,0,5,1

%N Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -25.

%H Amiram Eldar, <a href="/A035165/b035165.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Amiram Eldar_, Nov 17 2023: (Start)

%F a(n) = Sum_{d|n} Kronecker(-25, d).

%F Multiplicative with a(5^e) = 1, a(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4) (p is in A002145), and a(p^e) = e+1 if p = 2 or p > 5 and p == 1 (mod 4) (p = 2 or p is in A002144 \ {5}).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/5 = 1.256637... (A019694). (End)

%t a[n_] := DivisorSum[n, KroneckerSymbol[-25, #] &]; Array[a, 100] (* _Amiram Eldar_, Nov 17 2023 *)

%o (PARI) my(m = -25); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

%o (PARI) a(n) = sumdiv(n, d, kronecker(-25, d)); \\ _Amiram Eldar_, Nov 17 2023

%Y Cf. A002144, A002145, A019694.

%K nonn,easy,mult

%O 1,2

%A _N. J. A. Sloane_.