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With increasing n, a(n) goes to infinity (proof in Sierpiński).

From Bernard Schott, Oct 03 2022: (Start)

It seems that sigma(n!)/n! is an integer only for n = 0, 1, 3, 5 and corresponding values are 1, 1, 2, 3.

For m >= 2, the smallest integer n such that a(n) = m is A061556(m). (End)

Theorem: Let p(n) be the smallest prime such that Product_{prime p<=p(n)} 1/(1-1/p) >= n. Then lim_{n->oo} p(n+1)/p(n) = exp(exp(-gamma)).

Proof. Follow Mertens's Third Theorem Product_{p<=x} 1/(1-1/p) ~ log(x)/exp(-gamma).

For any particular integer n, it follows from the equations n = log(x_n)/exp(-gamma) -> x_n = exp(n*exp(-gamma)) and n+1 = log(x_n+1)/exp(-gamma) -> x_n+1 = exp((n+1)*exp(-gamma)) that lim_{n->oo} exp((n+1)*exp(-gamma))/exp((n)*exp(-gamma)) = exp(exp(-gamma)).

Convergence table:

n p(n) truncated Euler product up to p(n) ratio p(n)/p(n-1)

42 17427088769 42.0000000010939727723681242652955 1.7532416978341651

43 30553756811 43.0000000012946363551468233325186 1.7532335558736718

44 53567706007 44.0000000002803554088007272169139 1.7532281329055578

45 93916601047 45.0000000002681963271546340553884 1.7532317145469581

46 164657625967 46.0000000002470257389410099668348 1.7532323799133028

47 288682860119 47.0000000001305074313442036255929 1.7532310357544971

48 506127311983 48.0000000000705764045487316221655 1.7532295189758258

oo oo oo 1.7532294434956945