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A295629 Number of partitions of n into two parts such that not both are prime. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

LINKS

Table of n, a(n) for n=1..70.

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}]

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

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Last modified February 26 17:19 EST 2020. Contains 332293 sequences. (Running on oeis4.)