A307726 -id:A307726 - 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 *)

A307727 Number of partitions of n into 3 prime powers (not including 1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 3, 4, 5, 6, 6, 8, 7, 9, 9, 10, 10, 12, 11, 14, 13, 14, 13, 16, 13, 18, 15, 18, 16, 20, 18, 23, 20, 25, 23, 26, 22, 28, 23, 30, 23, 30, 23, 32, 26, 32, 27, 34, 28, 37, 28, 36, 29, 40, 31, 43, 28, 42, 32, 44, 32, 46, 32, 46, 35, 46, 35, 50, 34, 51, 37, 53, 36, 59, 36, 57, 41

Offset: 0

Ilya Gutkovskiy, Apr 24 2019

			a(11) = 4 because we have [7, 2, 2], [5, 4, 2], [5, 3, 3] and [4, 4, 3].

Robert Israel, Table of n, a(n) for n = 0..1000
Index entries for sequences related to partitions

Cf. A000961, A023894, A068307, A246655, A280243, A307726.

f:= proc(n,k,pmax) option remember;
  local t,p,j;
  if n = 0 then return `if`(k=0, 1, 0) fi;
  if k = 0 then return 0 fi;
  if n > k*pmax then return 0 fi;
  t:= 0:
  for p in A246655 do
    if p > pmax then return t fi;
    t:= t + add(procname(n-j*p, k-j, min(p-1,n-j*p)),j=1..min(k,floor(n/p)))
  od;
  t
end proc:
seq(f(n,3,n),n=0..80) # Robert Israel, Apr 25 2019

Array[Count[IntegerPartitions[#, {3}], _?(AllTrue[#, PrimePowerQ] &)] &, 81, 0]

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

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} [omega(i) * omega(j) * omega(n-i-j) == 1], where omega(n) is the number of distinct prime factors of n and [==] is the Iverson bracket. - Wesley Ivan Hurt, Apr 25 2019

A306433 Number of partitions of n into 2 distinct prime powers (not including 1).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 1, 2, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 4, 4, 3, 2, 4, 3, 3, 4, 4, 3, 5, 3, 5, 4, 6, 4, 7, 2, 4, 4, 6, 3, 5, 3, 5, 5, 5, 2, 7, 3, 6, 4, 6, 2, 7, 3, 7, 4, 5, 2, 7, 3, 5, 4, 6, 2, 9, 2, 7, 5, 7, 2, 9, 3, 6, 6, 7, 3, 9, 2, 8, 4, 5, 4, 10, 3, 8, 4, 7, 3, 11, 4, 8, 3, 6, 2

Offset: 0

Ilya Gutkovskiy, Apr 30 2019

nonn

			a(12) = 3 because we have [9, 3], [8, 4] and [7, 5].

Index entries for sequences related to partitions

Cf. A000961, A071330, A106244, A117929, A246655, A307726, A307825.

Table[Count[IntegerPartitions[n, {2}], _?(And[UnsameQ @@ #, AllTrue[#, PrimePowerQ[#] &]] &)], {n, 0, 95}]

a(n) = [x^n y^2] 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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A307727 Number of partitions of n into 3 prime powers (not including 1).

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A306433 Number of partitions of n into 2 distinct prime powers (not including 1).

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