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

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

Author

Ilya Gutkovskiy, Apr 24 2019

Keywords

• nonn
• look

Examples

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

Links

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

Crossrefs

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

Programs

Maple

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

Mathematica

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

Formula

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