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A034891
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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.
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40
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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
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OFFSET
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0,3
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COMMENTS
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Number of partitions of n into noncomposite parts. - Omar E. Pol, Jun 23 2022
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LINKS
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FORMULA
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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
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MAPLE
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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)):
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
(Sage) [Partitions(n, parts_in=(prime_range(n+1)+[1])).cardinality() for n in xsrange(1000)] # Giuseppe Coppoletta, Jul 11 2016
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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