A339708 a(n) is the number of decompositions of 2*n as the sum of an odd prime and a semiprime.

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 1, 3, 4, 1, 4, 2, 2, 6, 4, 3, 5, 5, 2, 4, 7, 4, 7, 6, 3, 10, 5, 4, 10, 6, 6, 7, 8, 5, 9, 9, 4, 8, 10, 4, 11, 10, 9, 13, 9, 7, 10, 10, 9, 10, 9, 8, 11, 13, 4, 16, 13, 9, 15, 11, 11, 13, 14, 13, 13, 10, 10, 15, 16, 8, 19, 11, 11, 17, 14, 15, 17, 18, 9, 13, 17, 15

Offset: 1

Author

J. M. Bergot and Robert Israel, Dec 13 2020

Keywords

• nonn
• look

Examples

			a(10) = 2 because 20 = 5+15 = 11+9 where 5 and 11 are primes and 15 and 9 are semiprimes.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A001358, A235645, A339709.

Programs

Maple

N:= 300: # for a(1)..a(N/2)
P:= select(isprime,[seq(i,i=3..N,2)]):
S:= sort(select(`<`,[seq(seq(P[i]*P[j],i=1..j),j=1..nops(P))],N)):
V:= Vector(N):
for p in P do
for s in S do
  v:= p+s;
    if v>N then break fi;
  V[v]:= V[v]+1
od od:
seq(V[i],i=2..N,2);

Mathematica

{0}~Join~Array[Count[IntegerPartitions[2 #, {2}, All, -(# - 2)], ?(And[AnyTrue[#, PrimeQ], AnyTrue[#, PrimeOmega[#] == 2 &]] &)] &, 86, 2] (* _Michael De Vlieger, Dec 13 2020 *)

PARI

a(n) = {my(nb=0); forprime(p=3, 2*n, if (bigomega(2*n-p) == 2, nb++);); nb;} \\ Michel Marcus, Dec 14 2020