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

1,10

Comments

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

Links

Table of n, a(n) for n=1..82.

Formula

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A000040, A000586, A010051, A166081.

Sequence in context: A174007 A330709 A045450 * A029222 A162350 A334305

Adjacent sequences: A347416 A347417 A347418 * A347420 A347421 A347422

Keyword

nonn

Author

Ayoub Saber Rguez, Aug 31 2021

Status

approved