A066843 a(n) = Product_{k=1..n} d(k); d(k) = A000005(k) is the number of positive divisors of k.

1, 1, 2, 4, 12, 24, 96, 192, 768, 2304, 9216, 18432, 110592, 221184, 884736, 3538944, 17694720, 35389440, 212336640, 424673280, 2548039680, 10192158720, 40768634880, 81537269760, 652298158080, 1956894474240, 7827577896960, 31310311587840, 187861869527040

Offset: 0

Author

Leroy Quet, Jan 20 2002

Keywords

• nonn

Comments

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = d_3(gcd(i,j)) for 1 <= i,j <= n, where d_3(n) is A007425. - Enrique Pérez Herrero, Aug 12 2011

a(n) is the number of integer sequences of length n where a(m) divides m for every term. - Franklin T. Adams-Watters, Oct 29 2017

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1310 (terms n = 1..200 from Harry J. Smith)
• Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49
• Mathoverflow, Product of tau(k), 2015.
• Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (10).

Crossrefs

Cf. A000005, A001088, A066780.

Programs

Maple

with(numtheory):seq(mul(tau(k),k=1..n), n=0..26); # Zerinvary Lajos, Jan 11 2009
with(numtheory):a[0]:=1: for n from 2 to 26 do a[n]:=a[n-1]*tau(n) od: seq(a[n], n=0..26); # Zerinvary Lajos, Mar 21 2009

Mathematica

A066843[n_] := Product[DivisorSigma[0,i], {i,1,n}]; Array[A066843,20] (* Enrique Pérez Herrero, Aug 12 2011 *)
FoldList[Times, Array[DivisorSigma[0, #] &, 27]] (* Michael De Vlieger, Nov 01 2017 *)

PARI

{ p=1; for (n=1, 200, p*=length(divisors(n)); write("b066843.txt", n, " ", p) ) } \\ Harry J. Smith, Apr 01 2010

Formula

a(n) = Product_{p=primes<=n} Product_{1<=k<=log(n)/log(p)} (1 +1/k)^floor(n/p^k). - Leroy Quet, Mar 20 2007

a(n) = Product_{k=1..n} Product_{p prime<=n} (v_p(k) + 1), where v_p(k) is the exponent of highest power of p dividing k. - Ridouane Oudra, Apr 15 2024

Extensions

a(0)=1 prepended by Alois P. Heinz, Jul 19 2023