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G.f.: exp( Sum_{n>=1} sigma(n^5)*x^n/n ).
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%I #18 Nov 01 2024 11:09:47

%S 1,1,32,153,1145,5677,37641,184685,1047862,5196410,26935148,129702476,

%T 638028933,2987297287,14055935617,64139004752,291595380989,

%U 1296984485909,5732084828019,24910785830408,107411267744602,457008372687439,1928413165110846,8046605441623654

%N G.f.: exp( Sum_{n>=1} sigma(n^5)*x^n/n ).

%C In general, if m >= 1 and g.f.= exp(Sum_{k>=1} sigma(k^m)*x^k/k), then log(a(n)) ~ (1 + 1/m) * (c*m!)^(1/(m+1)) * n^(m/(m+1)), where c = Product_{primes p} ((p^(m+2) - p^(m+1) + p^m - p) / ((p-1)*(p^(m+1)-1))). - _Vaclav Kotesovec_, Nov 01 2024

%H Seiichi Manyama, <a href="/A203557/b203557.txt">Table of n, a(n) for n = 0..1000</a>

%F Logarithmic derivative yields A203556.

%F a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A203556(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Sep 09 2020

%F log(a(n)) ~ 2^(3/2) * 3^(7/6) * c^(1/6) * n^(5/6) / 5^(5/6), where c = Product_{primes p} (p*(1 + p + p^2 + p^3 + p^5) / (p^6 - 1)) = 1.93252811194652723494722635658171746713... - _Vaclav Kotesovec_, Nov 01 2024

%e G.f.: A(x) = 1 + x + 32*x^2 + 153*x^3 + 1145*x^4 + 5677*x^5 + 37641*x^6 +...

%e where the logarithm equals the l.g.f. of A203556:

%e log(A(x)) = x + 63/2*x^2 + 364/3*x^3 + 2047/4*x^4 + 3906/5*x^5 +...+ sigma(n^5)*x^n/n +...

%o (PARI) {a(n)=polcoeff(exp(sum(m=1,n,sigma(m^5)*x^m/m)+x*O(x^n)),n)}

%Y Cf. A203556, A000203 (sigma); variants: A000041 (m=1), A156303 (m=2), A156304 (m=3), A202993 (m=4).

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 03 2012