A331923 Number of compositions (ordered partitions) of n into distinct perfect powers.
1, 1, 0, 0, 1, 2, 0, 0, 1, 3, 2, 0, 2, 8, 6, 0, 1, 4, 6, 0, 2, 12, 24, 0, 2, 9, 8, 1, 8, 32, 30, 2, 7, 10, 32, 8, 11, 44, 150, 30, 34, 40, 18, 26, 20, 68, 78, 126, 56, 169, 80, 30, 40, 116, 294, 144, 162, 226, 182, 128, 66, 338, 348, 752, 199, 1048
Offset: 0
Examples
a(17) = 4 because we have [16, 1], [9, 8], [8, 9] and [1, 16].
Links
Programs
-
Maple
N:= 200: # for a(0)..a(N) PP:= {1,seq(seq(b^i,i=2..floor(log[b](N))),b=2..floor(sqrt(N)))}: G:= mul(1+t*x^p, p=PP): F:= proc(n) local R, k, v; R:= normal(coeff(G, x, n)); add(k!*coeff(R, t, k), k=1..degree(R, t)) end proc: F(0):= 1: map(F, [$0..N]); # Robert Israel, Feb 03 2020 -
Mathematica
M = 200; PP = Join[{1}, Table[Table[b^i, {i, 2, Floor[Log[b, M]]}], {b, 2, Floor[ Sqrt[M]]}] // Flatten // Union]; G = Product[1 + t x^p, {p, PP}]; a[n_] := Module[{R, k, v}, R = SeriesCoefficient[G, {x, 0, n}]; Sum[k! SeriesCoefficient[R, {t, 0, k}], {k, 1, Exponent[R, t]}]]; a[0] = 1; a /@ Range[0, M] (* Jean-François Alcover, Oct 25 2020, after Robert Israel *)