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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.

%I A066843 #60 May 12 2024 11:34:55
%S A066843 1,1,2,4,12,24,96,192,768,2304,9216,18432,110592,221184,884736,
%T A066843 3538944,17694720,35389440,212336640,424673280,2548039680,10192158720,
%U A066843 40768634880,81537269760,652298158080,1956894474240,7827577896960,31310311587840,187861869527040
%N A066843 a(n) = Product_{k=1..n} d(k); d(k) = A000005(k) is the number of positive divisors of k.
%C A066843 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
%C A066843 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
%H A066843 Alois P. Heinz, <a href="/A066843/b066843.txt">Table of n, a(n) for n = 0..1310</a> (terms n = 1..200 from Harry J. Smith)
%H A066843 Antal Bege, <a href="http://www.emis.de/journals/AUSM/C1-1/MATH1-4.PDF">Hadamard product of GCD matrices</a>, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49
%H A066843 Mathoverflow, <a href="https://mathoverflow.net/questions/208827/product-of-tauk">Product of tau(k)</a>, 2015.
%H A066843 Ramanujan's Papers, <a href="https://web.archive.org/web/20200124035942/http://ramanujan.sirinudi.org/Volumes/published/ram17.html">Some formulas in the analytic theory of numbers</a>, Messenger of Mathematics, XLV, 1916, 81-84, Formula (10).
%F A066843 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
%F A066843 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
%p A066843 with(numtheory):seq(mul(tau(k),k=1..n), n=0..26); # _Zerinvary Lajos_, Jan 11 2009
%p A066843 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
%t A066843 A066843[n_] := Product[DivisorSigma[0,i], {i,1,n}]; Array[A066843,20] (* _Enrique Pérez Herrero_, Aug 12 2011 *)
%t A066843 FoldList[Times, Array[DivisorSigma[0, #] &, 27]] (* _Michael De Vlieger_, Nov 01 2017 *)
%o A066843 (PARI) { p=1; for (n=1, 200, p*=length(divisors(n)); write("b066843.txt", n, " ", p) ) } \\ _Harry J. Smith_, Apr 01 2010
%Y A066843 Cf. A000005, A001088, A066780.
%K A066843 nonn
%O A066843 0,3
%A A066843 _Leroy Quet_, Jan 20 2002
%E A066843 a(0)=1 prepended by _Alois P. Heinz_, Jul 19 2023