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A182270
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Number of representations of n as a sum of products of pairs of integers larger than 1, considered to be equivalent when terms or factors are reordered.
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14
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1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 0, 5, 1, 4, 2, 9, 2, 11, 3, 16, 7, 19, 6, 34, 13, 35, 18, 57, 23, 73, 32, 99, 53, 125, 60, 186, 92, 215, 127, 311, 164, 394, 221, 518, 320, 656, 386, 903, 545, 1091, 719, 1470, 925, 1863, 1215, 2390, 1642, 3015, 2037, 3966
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OFFSET
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0,9
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LINKS
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FORMULA
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G.f.: Product_{k>0} 1/(1-x^k)^(A038548(k)-1).
G.f.: Product_{i>=1} Product_{j=2..i} 1/(1 - x^(i*j)). - Ilya Gutkovskiy, Sep 23 2019
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EXAMPLE
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a(0) = 1: 0 = the empty sum.
a(1) = a(2) = a(3) = 0: no product is < 4.
a(4) = 1: 4 = 2*2.
a(6) = 1: 6 = 2*3.
a(8) = 2: 8 = 2*2 + 2*2 = 2*4.
a(9) = 1: 9 = 3*3.
a(12) = 5: 12 = 2*2 + 2*2 + 2*2 = 2*2 + 2*4 = 2*3 + 2*3 = 2*6 = 3*4.
a(13) = 1: 13 = 2*2 + 3*3.
a(14) = 4: 14 = 2*2 + 2*2 + 2*3 = 2*3 + 2*4 = 2*2 + 2*5 = 2*7.
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MAPLE
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with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
d*(ceil(tau(d)/2)-1), d=divisors(j)) *a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..70);
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MATHEMATICA
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a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*(Ceiling[DivisorSigma[0, d]/2] - 1), {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Sep 09 2014, after Alois P. Heinz *)
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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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