_Paul D. Hanna (pauldhanna(AT)juno.com), _, Sep 01 2009

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

1, 1, 2, 7, 41, 385, 5769, 139541, 5551356, 369312953, 41588540350, 7987225089655, 2629160183190431, 1487755631073862696, 1450453417949809255147, 2439516473122553169216351, 7086426394313598512496200542

0,3

Conjecture: if F(x) = exp( Sum_{n>=1} L(n)*x^n/n ) is an integer series,

then the g.f. that satisfies:

G(x) = exp( Sum_{n>=1} G( L(n)*x )^n*x^n/n ) is also an integer series.

Another example of this is A157675 in which L(n) = 2^n.

G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 41*x^4 + 385*x^5 + 5769*x^6 +...

log(A(x)) = A(x)*x + A(3x)^2*x^2/2 + A(4x)^3*x^3/3 + A(7x)^4*x^4/4 +...

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(k=1, n, subst(A, x, sigma(k)*x+x*O(x^n))^k*x^k/k))); polcoeff(A, n)}

Cf. A157675 (variant), A000203 (sigma).

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Sep 01 2009

approved