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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. 13
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n) = length of n-th row in A212721. - Reinhard Zumkeller, Jun 14 2012 LINKS T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n=1..1000 from T. D. Noe) 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 MAPLE b:= proc(n, i) option remember; local p;       p:= `if`(i<1, 1, ithprime(i));       `if`(n=0, 1, `if`(i<0, 0, b(n, i-1)+       `if`(p>n, 0, b(n-p, 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 *) PROG (Haskell) a034891 = length . a212721_row  -- Reinhard Zumkeller, Jun 14 2012 (Sage) [Partitions(n, parts_in=(prime_range(n+1)+[1])).cardinality() for n in xsrange(1000)] # Giuseppe Coppoletta, Jul 11 2016 CROSSREFS Cf. A000792, A000793, A009490. Cf. A140436. Sequence in context: A114829 A175869 A007279 * A143611 A279075 A062464 Adjacent sequences:  A034888 A034889 A034890 * A034892 A034893 A034894 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS More terms from Vladeta Jovovic a(0)=1 from Michael Somos, Feb 05 2011 STATUS approved

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