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 A034891 Number of different products of partitions of n; number of partitions of n into prime parts (1 included); number of distinct orders of Abelian subgroups of symmetric group S_n. 10

%I

%S 1,1,2,3,4,6,8,11,14,18,23,29,36,45,55,67,81,98,117,140,166,196,231,

%T 271,317,369,429,496,573,660,758,869,993,1133,1290,1465,1662,1881,

%U 2125,2397,2699,3035,3407,3820,4276,4780,5337,5951,6628,7372,8191,9090

%N Number of different products of partitions of n; number of partitions of n into prime parts (1 included); number of distinct orders of Abelian subgroups of symmetric group S_n.

%C a(n) = length of n-th row in A212721. - _Reinhard Zumkeller_, Jun 14 2012

%H T. D. Noe, <a href="/A034891/b034891.txt">Table of n, a(n) for n=1..1000</a>

%F G.f.: (1/(1-x))*(1/Product_{k>0} (1-x^prime(k))). a(n) = (1/n)*Sum_{k=1..n} A074372(k)*a(n-k). Partial sums of A000607. - _Vladeta Jovovic_, Sep 19 2002

%p b:= proc(n, i) option remember; local p;

%p p:= `if`(i<1, 1, ithprime(i));

%p `if`(n=0, 1, `if`(i<0, 0, b(n, i-1)+

%p `if`(p>n, 0, b(n-p, i))))

%p end:

%p a:= n-> b(n, numtheory[pi](n)):

%p seq(a(n), n=1..100); # _Alois P. Heinz_, Feb 15 2013

%t Table[ Length[ Union[ Apply[ Times, Partitions[ n ], 1 ] ] ], {n, 30} ]

%o a034891 = length . a212721_row -- _Reinhard Zumkeller_, Jun 14 2012

%Y Cf. A000792, A000793, A009490.

%K nonn,easy,nice,changed

%O 0,3

%A _Wouter Meeussen_