A302479 Number of partitions of n into two distinct nonprime parts.

0, 0, 0, 0, 1, 0, 1, 0, 1, 2, 1, 1, 2, 2, 2, 3, 2, 3, 3, 3, 3, 5, 3, 5, 4, 6, 4, 6, 5, 7, 6, 6, 6, 9, 6, 10, 7, 8, 8, 10, 8, 11, 9, 10, 9, 12, 9, 13, 10, 13, 10, 13, 11, 15, 12, 14, 12, 16, 13, 18, 14, 15, 14, 18, 14, 20, 15, 16, 16, 20, 16, 21, 17, 20, 17

Offset: 1

Author

Wesley Ivan Hurt, Apr 08 2018

Keywords

• nonn
• easy

Examples

			a(16) = 3; 16 = 15+1 = 12+4 = 10+6, which are distinct nonprimes.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to partitions

Crossrefs

Cf. A005171, A010051, A062610, A358638.

Cf. also A341461, A341462, A341464, A341465, A341466, A341467.

Programs

Mathematica

Table[Sum[(1 - PrimePi[n - i] + PrimePi[n - i - 1]) (1 - PrimePi[i] + PrimePi[i - 1]), {i, Floor[(n - 1)/2]}], {n, 100}]
Table[Length[Select[IntegerPartitions[n,{2}],Length[Union[#]]==2&&Boole[PrimeQ[#]]=={0,0}&]],{n,80}] (* Harvey P. Dale, Dec 28 2023 *)

PARI

A302479(n) = sum(k=1,(n-1)\2,!(isprime(k)+isprime(n-k))); \\ Antti Karttunen, Nov 25 2022

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} (1 - c(i)) * (1 - c(n-i)), where c = A010051.

For n > 0, a(n) = A358638(n) - A005171(n). - Antti Karttunen, Nov 25 2022