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.
1, 1, 2, 3, 4, 6, 8, 11, 14, 18, 23, 29, 36, 45, 55, 67, 81, 98, 117, 140, 166, 196, 231, 271, 317, 369, 429, 496, 573, 660, 758, 869, 993, 1133, 1290, 1465, 1662, 1881, 2125, 2397, 2699, 3035, 3407, 3820, 4276, 4780, 5337, 5951, 6628, 7372, 8191, 9090
Offset: 0
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..1000 from T. D. Noe)
- Index entries for sequences related to partitions
Programs
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Haskell
a034891 = length . a212721_row -- Reinhard Zumkeller, Jun 14 2012
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Maple
b:= proc(n, i) option remember; `if`(n=0, 1, (p-> `if`(i<0, 0, b(n, i-1)+ `if`(p>n, 0, b(n-p, i))))(`if`(i<1, 1, ithprime(i)))) end: a:= n-> b(n, numtheory[pi](n)): seq(a(n), n=0..100); # Alois P. Heinz, Feb 15 2013 -
Mathematica
Table[ Length[ Union[ Apply[ Times, Partitions[ n], 1]]], {n, 30}] CoefficientList[ Series[ (1/(1 - x)) Product[1/(1 - x^Prime[i]), {i, 100}], {x, 0, 50}], x] (* Robert G. Wilson v, Aug 17 2013 *) b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n == 0, 1, If[i<0, 0, b[n, i-1] + If[p>n, 0, b[n-p, i]]]]]; a[n_] := b[n, PrimePi[n] ]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 05 2015, after Alois P. Heinz *) -
Sage
[Partitions(n, parts_in=(prime_range(n+1)+[1])).cardinality() for n in xsrange(1000)] # Giuseppe Coppoletta, Jul 11 2016
Formula
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
Extensions
More terms from Vladeta Jovovic
a(0)=1 from Michael Somos, Feb 05 2011
Comments