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A047968
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a(n) = Sum_{d|n} p(d), where p(d) = A000041 = number of partitions of d.
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57
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1, 3, 4, 8, 8, 17, 16, 30, 34, 52, 57, 99, 102, 153, 187, 261, 298, 432, 491, 684, 811, 1061, 1256, 1696, 1966, 2540, 3044, 3876, 4566, 5846, 6843, 8610, 10203, 12610, 14906, 18491, 21638, 26508, 31290, 38044, 44584, 54133, 63262, 76241
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OFFSET
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1,2
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COMMENTS
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Inverse Moebius transform of A000041.
Sum of the partition numbers of the divisors of n. - Omar E. Pol, Feb 25 2014
Number of constant multiset partitions of multisets spanning an initial interval of positive integers with multiplicities an integer partition of n. - Gus Wiseman, Sep 16 2018
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LINKS
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FORMULA
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G.f.: Sum_{k>0} (-1+1/Product_{i>0} (1-z^(k*i))). - Vladeta Jovovic, Jun 22 2003
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EXAMPLE
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For n = 10 the divisors of 10 are 1, 2, 5, 10, hence the partition numbers of the divisors of 10 are 1, 2, 7, 42, so a(10) = 1 + 2 + 7 + 42 = 52. - Omar E. Pol, Feb 26 2014
The a(6) = 17 constant multiset partitions:
(111111) (111)(111) (11)(11)(11) (1)(1)(1)(1)(1)(1)
(111222) (12)(12)(12)
(111122) (112)(112)
(112233) (123)(123)
(111112)
(111123)
(111223)
(111234)
(112234)
(112345)
(123456)
(End)
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MAPLE
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with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l) do c := c+numbpart(l[i]) od: RETURN(c): end: for j from 1 to 60 do printf(`%d, `, a(j)) od: # Zerinvary Lajos, Apr 14 2007
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MATHEMATICA
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a[n_] := Sum[ PartitionsP[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 44}] (* Jean-François Alcover, Oct 03 2013 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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