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%I #55 Mar 27 2022 03:15:13
%S 1,3,12,84,504,6048,48384,725760,9434880,169827840,2037934080,
%T 57062154240,798870159360,19172883824640,460149211791360,
%U 14264625565532160,256763260179578880,10013767147003576320,200275342940071526400,8411564403483004108800
%N a(n) = Product_{k=1..n} sigma(k); sigma(k) is the sum of the positive divisors of n.
%C a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = A007429(gcd(i,j)) for 1 <= i,j <= n. - _Enrique Pérez Herrero_, Aug 12 2011
%H Paul D. Hanna, <a href="/A066780/b066780.txt">Table of n, a(n) for n = 1..300</a> (terms 1..100 from _Harry J. Smith_).
%H Antal Bege, <a href="http://www.emis.de/journals/AUSM/C1-1/MATH1-4.PDF">Hadamard product of GCD matrices</a>, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49
%H Vaclav Kotesovec, <a href="/A066780/a066780.jpg">Plot of (a(n)^(1/n))/n for n = 1..10^7</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 Lim_{n->infinity} (a(n)^(1/n)) / n = A345144 / exp(1) = 0.57447937538407152396420163967936309825692994713661226083669171312803511135... - _Vaclav Kotesovec_, Jun 09 2021
%p with(numtheory):seq(mul(sigma(k),k=1..n), n=1..26); # _Zerinvary Lajos_, Jan 11 2009
%p with(numtheory):a[0]:=1: a[1]:=1: for n from 2 to 26 do a[n]:=a[n-1]*sigma(n) od: seq(a[n], n=0..18); # _Zerinvary Lajos_, Mar 21 2009
%t A066780[n_] := Product[DivisorSigma[1,i], {i,1,n}]; Array[A066780,20] (* _Enrique Pérez Herrero_, Aug 12 2011 *)
%t FoldList[Times,DivisorSigma[1,Range[20]]] (* _Harvey P. Dale_, Jan 29 2022 *)
%o (PARI) { p=1; for (n=1, 100, write("b066780.txt", n, " ", p*=sigma(n)) ) } \\ _Harry J. Smith_, Mar 25 2010
%Y Cf. A000203, A001088, A066843, A294343, A345144.
%K nonn
%O 1,2
%A _Benoit Cloitre_ and _Leroy Quet_, Jan 18 2002
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