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%I #12 Aug 24 2023 02:31:16
%S 2,8,536,64960,33554592,68718964352,562949953422208,
%T 18446744065119352832,2417851639229258752070144,
%U 1267650600228227149696894752768,2658455991569831745807614120560711680,22300745198530623141526273540526772167065600,748288838313422294120286634350736906063837462110208,100433627766186892221372630770688837357523572410678079422464
%N a(n) = Sum_{d|n} -(-1)^d * 2^(n^2/d) * d.
%C Equals the logarithmic derivative of A158096.
%F L.g.f.: Sum_{n>=1} x^n/n * 2^(n^2)/(1 + 2^(n^2)*x^n).
%F Conjecture: a(n) = 2^A007814(n) * 2^n * d for some odd d, where A007814(n) equals the exponent of highest power of 2 dividing n.
%e L.g.f.: L(x) = 2*x + 8*x^2/2 + 536*x^3/3 + 64960*x^4/4 + 33554592*x^5/5 + ...
%e where
%e L(x) = 2/(1 + 2*x)*x + 2^4/(1 + 2^4*x^2)*x^2/2 + 2^9/(1 + 2^9*x^3)*x^3/3 + 2^16/(1 + 2^16*x^4)*x^4/4 + 2^25/(1 + 2^25*x^5)*x^5/5 + ...
%e and
%e exp(L(x)) = 1 + 2*x + 6*x^2 + 188*x^3 + 16614*x^4 + 6744492*x^5 + 11466697660*x^6 + 80444371592472*x^7 + ...+ A158096(n)*x^n + ...
%t a[n_] := DivisorSum[n, -(-1)^# * 2^(n^2/#) * # &]; Array[a, 14] (* _Amiram Eldar_, Aug 24 2023 *)
%o (PARI) {a(n) = n*polcoeff(sum(k=1, n, x^k/k * 2^(k^2)/(1 + 2^(k^2)*x^k +x*O(x^n))), n)}
%o for(n=1,20,print1(a(n),", "))
%o (PARI) {a(n) = sumdiv(n, d, -(-1)^d * 2^(n^2/d) * d)}
%o for(n=1,20,print1(a(n),", "))
%Y Cf. A007814, A158096 (exp).
%K nonn
%O 1,1
%A _Paul D. Hanna_, Oct 03 2015
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