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a(n) = Product_{i=1..n} sigma_2(i).
3

%I #34 Mar 27 2022 03:15:47

%S 1,5,50,1050,27300,1365000,68250000,5801250000,527913750000,

%T 68628787500000,8372712075000000,1758269535750000000,

%U 298905821077500000000,74726455269375000000000,19428878370037500000000000,6625247524182787500000000000

%N a(n) = Product_{i=1..n} sigma_2(i).

%C a(n) is the product of the sum of the squared divisors of i, for i from 1 to n.

%H Vaclav Kotesovec, <a href="/A247951/b247951.txt">Table of n, a(n) for n = 1..248</a>

%H Vaclav Kotesovec, <a href="/A247951/a247951.jpg">Plot of (a(n)/(n!)^2)^(1/n) for n = 1..100000</a>

%H Ramanujan's Papers, <a href="https://web.archive.org/web/20200124035942/http://ramanujan.sirinudi.org/Volumes/published/ram17.html">Some formulas in the analytic theory of numbers</a>, Messenger of Mathematics, XLV, 1916, 81-84, Formula (20).

%F a(n) = Product_{i=1..n} A001157(i).

%F Lim_{n->infinity} (a(n) / (n!)^2)^(1/n) = A345158. - _Vaclav Kotesovec_, Jun 10 2021

%p with(numtheory): A247951:=n->mul(sigma[2](i),i=1..n): seq(A247951(n), n=1..20);

%t Table[Product[DivisorSigma[2, i], {i, n}], {n, 20}]

%o (PARI) lista(nn) = vector(nn, n, prod(i=1, n, sigma(i, 2))) \\ _Michel Marcus_, Oct 01 2014

%Y Cf. A000203 (sigma), A001157 (sigma_2), A066780 (product{i=1..n} sigma(i)), A066843, A345158, A345160.

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Oct 01 2014