A066780 a(n) = Product_{k=1..n} sigma(k); sigma(k) is the sum of the positive divisors of n.

1, 3, 12, 84, 504, 6048, 48384, 725760, 9434880, 169827840, 2037934080, 57062154240, 798870159360, 19172883824640, 460149211791360, 14264625565532160, 256763260179578880, 10013767147003576320, 200275342940071526400, 8411564403483004108800

Offset: 1

Author

Benoit Cloitre and Leroy Quet, Jan 18 2002

Keywords

• nonn

Comments

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

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..300 (terms 1..100 from _Harry J. Smith_).
• Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49
• Vaclav Kotesovec, Plot of (a(n)^(1/n))/n for n = 1..10^7
• Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (20).

Crossrefs

Cf. A000203, A001088, A066843, A294343, A345144.

Programs

Maple

with(numtheory):seq(mul(sigma(k),k=1..n), n=1..26); # Zerinvary Lajos, Jan 11 2009
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

Mathematica

A066780[n_] := Product[DivisorSigma[1,i], {i,1,n}]; Array[A066780,20] (* Enrique Pérez Herrero, Aug 12 2011 *)
FoldList[Times,DivisorSigma[1,Range[20]]] (* Harvey P. Dale, Jan 29 2022 *)

PARI

{ p=1; for (n=1, 100, write("b066780.txt", n, " ", p*=sigma(n)) ) } \\ Harry J. Smith, Mar 25 2010

Formula

Lim_{n->infinity} (a(n)^(1/n)) / n = A345144 / exp(1) = 0.57447937538407152396420163967936309825692994713661226083669171312803511135... - Vaclav Kotesovec, Jun 09 2021