A203557 G.f.: exp( Sum_{n>=1} sigma(n^5)*x^n/n ).

1, 1, 32, 153, 1145, 5677, 37641, 184685, 1047862, 5196410, 26935148, 129702476, 638028933, 2987297287, 14055935617, 64139004752, 291595380989, 1296984485909, 5732084828019, 24910785830408, 107411267744602, 457008372687439, 1928413165110846, 8046605441623654

Offset: 0

Author

Paul D. Hanna, Jan 03 2012

Keywords

• nonn

Comments

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

Examples

			G.f.: A(x) = 1 + x + 32*x^2 + 153*x^3 + 1145*x^4 + 5677*x^5 + 37641*x^6 +...
where the logarithm equals the l.g.f. of A203556:
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 +...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

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

Programs

PARI

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

Formula

Logarithmic derivative yields A203556.

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A203556(k)*a(n-k) for n > 0. - Seiichi Manyama, Sep 09 2020

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