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A066843 a(n) = Product_{k=1..n} d(k); d(k) = A000005(k) is the number of positive divisors of k. 18
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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).
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=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 *)
PROG
(PARI) { p=1; for (n=1, 200, p*=length(divisors(n)); write("b066843.txt", n, " ", p) ) } \\ Harry J. Smith, Apr 01 2010
CROSSREFS
Sequence in context: A367703 A320931 A096421 * A051905 A051426 A048148
KEYWORD
nonn
AUTHOR
Leroy Quet, Jan 20 2002
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Jul 19 2023
STATUS
approved

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Last modified February 21 09:51 EST 2024. Contains 370228 sequences. (Running on oeis4.)