A182270 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.
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
Offset: 0
Keywords
Examples
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.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..10000
- N. J. A. Sloane, Transforms
Programs
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Maple
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); -
Mathematica
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 *)
Formula
Euler transform of A038548-1.
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