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

1,6

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

Formula

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

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A004526, A010051, A061358.

Sequence in context: A175944 A063905 A130312 * A076272 A180101 A108035

Adjacent sequences: A295626 A295627 A295628 * A295630 A295631 A295632

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Nov 24 2017

Status

approved