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

%I A262009 #19 Aug 24 2023 02:31:19
%S A262009 2,24,530,65632,33554482,68719479000,562949953421410,
%T A262009 18446744073709814144,2417851639229258349417122,
%U A262009 1267650600228229401496837423704,2658455991569831745807614120560689394,22300745198530623141535718272648636384486240,748288838313422294120286634350736906063837462004050
%N A262009 Sum_{d|n} 2^(d^2) * n^2/d^2.
%C A262009 Logarithmic derivative of A262008.
%F A262009 a(n) = Sum_{d|n} 2^(n^2/d^2) * d^2.
%F A262009 a(2*n) == 0 (mod 8), a(2*n-1) == 2 (mod 8).
%F A262009 Conjecture: A037227(a(n)) = 2*A037227(n) + 1.
%F A262009 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.
%e A262009 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 + ...
%e A262009 where
%e A262009 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 + ...
%t A262009 a[n_] := DivisorSum[n, 2^(#^2) * (n/#)^2 &]; Array[a, 13] (* _Amiram Eldar_, Aug 24 2023 *)
%o A262009 (PARI) {a(n) = sumdiv(n,d, 2^(d^2) * n^2/d^2)}
%o A262009 for(n=1,20,print1(a(n),", "))
%Y A262009 Cf. A262008 (exp), A037227.
%K A262009 nonn
%O A262009 1,1
%A A262009 _Paul D. Hanna_, Oct 01 2015