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

1,2

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).

Formula

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A051549 A245775 A277782 * A328778 A231868 A147835

Adjacent sequences: A066777 A066778 A066779 * A066781 A066782 A066783

Keyword

nonn

Author

Benoit Cloitre and Leroy Quet, Jan 18 2002

Status

approved