A262009 Sum_{d|n} 2^(d^2) * n^2/d^2.

2, 24, 530, 65632, 33554482, 68719479000, 562949953421410, 18446744073709814144, 2417851639229258349417122, 1267650600228229401496837423704, 2658455991569831745807614120560689394, 22300745198530623141535718272648636384486240, 748288838313422294120286634350736906063837462004050

Offset: 1

Author

Paul D. Hanna, Oct 01 2015

Keywords

• nonn

Comments

Logarithmic derivative of A262008.

Examples

			L.g.f.: L(x) = 2*x + 24*x^2/2 + 530*x^3/3 + 65632*x^4/4 + 33554482*x^5/5 + 68719479000*x^6/6 + 562949953421410*x^7/7 + ...
where
exp(L(x)) = 1 + 2*x + 14*x^2 + 202*x^3 + 16858*x^4 + 6746346*x^5 + 11466918526*x^6 + ... + A262008(n)*x^n + ...
		

Crossrefs

Cf. A262008 (exp), A037227.

Programs

Mathematica

a[n_] := DivisorSum[n, 2^(#^2) * (n/#)^2 &]; Array[a, 13] (* Amiram Eldar, Aug 24 2023 *)

PARI

{a(n) = sumdiv(n,d, 2^(d^2) * n^2/d^2)}
for(n=1,20,print1(a(n),", "))

Formula

a(n) = Sum_{d|n} 2^(n^2/d^2) * d^2.

a(2*n) == 0 (mod 8), a(2*n-1) == 2 (mod 8).

Conjecture: A037227(a(n)) = 2*A037227(n) + 1.

Conjecture: a(n) = 2^A037227(n) * d for some odd d, where A037227(n) = 2*m + 1 such that n = 2^m * k for some odd k.