A347419 Number of partitions of n into two or more distinct primes.

0, 0, 0, 0, 1, 0, 1, 1, 1, 2, 0, 2, 1, 2, 2, 3, 1, 4, 2, 4, 4, 4, 4, 5, 5, 6, 5, 6, 6, 6, 8, 7, 9, 9, 9, 11, 10, 11, 13, 12, 13, 15, 14, 17, 16, 18, 18, 20, 21, 23, 22, 25, 25, 27, 30, 29, 32, 32, 34, 37, 38, 40, 42, 44, 45, 50, 49, 53, 55, 57, 60, 64, 66, 70, 71, 76, 78, 83, 86, 89, 93, 96

Offset: 1

Author

Ayoub Saber Rguez, Aug 31 2021

Keywords

• nonn

Comments

Every positive integer can be written as a sum of two or more distinct primes except 1,2,3,4,6 and 11.

Examples

			a(5) = 1: 2+3.
a(18) = 4: 11+7, 11+5+2, 13+5, 13+3+2.
		

Crossrefs

Cf. A000040, A000586, A010051, A166081.

Programs

Maple

h:= proc(n) h(n):=`if`(n<2, 0, `if`(isprime(n), n, h(n-1))) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,
      b(n, h(i-1))+b(n-i, h(min(n-i, i-1)))))
    end:
a:= n-> b(n, h(n-1)):
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 03 2021

Mathematica

m = 24; Rest @ CoefficientList[Series[Product[(1 + x^Prime[k]), {k, 1, m}], {x, 0, Prime[m]}], x] - Table[Boole @ PrimeQ[n], {n, 1, Prime[m]}] (* Amiram Eldar, Sep 03 2021 *)

Python

from sympy import isprime, primerange
from functools import cache
@cache
def A000586(n, k=None): # after Charles R Greathouse IV
    if k == None: k = n
    if n < 1: return int(n == 0)
    return sum(A000586(n-p, p-1) for p in primerange(1, k+1))
def a(n): return A000586(n) - isprime(n)
print([a(n) for n in range(1, 83)]) # Michael S. Branicky, Sep 03 2021

Formula

a(n) = A000586(n) - A010051(n).