A306433 -id:A306433 - cp's OEIS frontend

A341132 Number of partitions of n into 2 distinct prime powers (including 1).

1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 5, 4, 3, 2, 5, 3, 4, 4, 5, 3, 6, 3, 6, 5, 6, 4, 7, 2, 5, 4, 6, 3, 6, 3, 6, 5, 5, 2, 8, 3, 7, 4, 6, 2, 8, 3, 7, 4, 5, 2, 8, 3, 6, 4, 6, 3, 9, 2, 8, 5, 7, 2, 10, 3, 7, 6, 7, 3, 9, 2, 9, 4, 6, 4, 11, 3, 8, 4, 7, 3, 12

Offset: 3

Ilya Gutkovskiy, Feb 05 2021

nonn

Cf. A000961, A010055, A071330, A282062, A306433, A307726, A341140, A341141, A341142, A341143, A341144, A341145.

q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(q(i), b(n-i, min(n-i, i-1), t-1), 0)))
    end:
a:= n-> b(n$2, 2):
seq(a(n), n=3..90);  # Alois P. Heinz, Feb 05 2021

q[n_] := q[n] = PrimeNu[n] < 2;
b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[q[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 2];
Table[a[n], {n, 3, 90}] (* Jean-François Alcover, Jul 13 2021, after Alois P. Heinz *)

A307825 Number of partitions of n into 3 distinct prime powers (not including 1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 4, 4, 5, 4, 6, 5, 7, 6, 8, 8, 10, 8, 10, 9, 12, 11, 12, 11, 15, 12, 15, 14, 17, 17, 20, 18, 19, 19, 19, 22, 23, 20, 21, 24, 23, 24, 24, 24, 27, 28, 24, 27, 28, 28, 28, 33, 27, 33, 29, 31, 30, 35, 27, 35, 33, 34, 31, 40, 32, 42, 35, 39, 35, 47, 32

Offset: 0

Ilya Gutkovskiy, Apr 30 2019

nonn

			a(15) = 4 because we have [9, 4, 2], [8, 5, 2], [8, 4, 3] and [7, 5, 3].

Index entries for sequences related to partitions

Cf. A000961, A106244, A125688, A246655, A306433, A307727.

Table[Count[IntegerPartitions[n, {3}], _?(And[UnsameQ @@ #, AllTrue[#, PrimePowerQ[#] &]] &)], {n, 0, 78}]

a(n) = [x^n y^3] Product_{k>=1} (1 + y*x^A246655(k)).

cp's OEIS Frontend

A341132 Number of partitions of n into 2 distinct prime powers (including 1).

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