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A066843 a(n) = Product_{k=1..n} d(k); d(k) is the number of positive divisors of k. 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

Harry J. Smith, Table of n, a(n) for n = 1..200

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).

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

MAPLE

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

with(numtheory):a[1]:=1: for n from 2 to 26 do a[n]:=a[n-1]*tau(n) od: seq(a[n], n=1..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 *)

PROG

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

CROSSREFS

Cf. A000005, A001088, A066780.

Sequence in context: A200337 A320931 A096421 * A051905 A051426 A048148

Adjacent sequences:  A066840 A066841 A066842 * A066844 A066845 A066846

KEYWORD

nonn,changed

AUTHOR

Leroy Quet, Jan 20 2002

STATUS

approved

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Last modified June 16 04:31 EDT 2021. Contains 345055 sequences. (Running on oeis4.)