A295629 Number of partitions of n into two parts such that not both are prime.

0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 6, 6, 8, 7, 8, 8, 9, 8, 11, 9, 11, 10, 13, 12, 14, 12, 14, 14, 15, 13, 17, 14, 18, 17, 18, 17, 20, 17, 20, 19, 21, 19, 23, 19, 23, 21, 25, 23, 26, 22, 26, 25, 28, 25, 29, 24, 29, 28, 30, 27, 32, 27, 33, 32, 33, 30

Offset: 1

Author

Wesley Ivan Hurt, Nov 24 2017

Keywords

• nonn
• easy

Examples

			a(8) = 3; the partitions of 8 into two parts are (7,1), (6,2), (5,3) and (4,4). Since the parts in (7,1), (6,2) and (4,4) are not both prime, a(8) = 3.
a(11) = 5; the partitions of 11 into two parts are (10,1), (9,2), (8,3), (7,4) and (6,5). All of these have parts that are not both prime, so a(11) = 5.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to Goldbach conjecture
• Index entries for sequences related to partitions

Crossrefs

Cf. A004526, A010051, A061358.

Programs

Maple

N:= 1000: # to get a(1)..a(N)
P:= select(isprime, [2,seq(i,i=3..N,2)]):
A:= Vector(N,t -> floor(t/2)):
for i from 1 to nops(P) do
  for j from i to nops(P) do
    m:= P[i]+P[j];
    if m > N then break fi;
    A[m]:= A[m]-1;
od od:
convert(A,list); # Robert Israel, Dec 07 2017

Mathematica

Table[Sum[1 - (PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i] - PrimePi[n - i - 1]), {i, Floor[n/2]}], {n, 80}]
Table[Total[If[AllTrue[#,PrimeQ],0,1]&/@IntegerPartitions[n,{2}]],{n,70}] (* Harvey P. Dale, Jan 17 2024 *)

PARI

a(n) = sum(i=1, floor(n/2), 1 - isprime(i)*isprime(n-i)) \\ Iain Fox, Dec 06 2017

Formula

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

a(n) = A004526(n) - A061358(n).