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

%I A339708 #19 Dec 15 2020 18:30:32
%S A339708 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,
%T A339708 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,
%U A339708 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
%N A339708 a(n) is the number of decompositions of 2*n as the sum of an odd prime and a semiprime.
%H A339708 Robert Israel, <a href="/A339708/b339708.txt">Table of n, a(n) for n = 1..5000</a>
%e A339708 a(10) = 2 because 20 = 5+15 = 11+9 where 5 and 11 are primes and 15 and 9 are semiprimes.
%p A339708 N:= 300: # for a(1)..a(N/2)
%p A339708 P:= select(isprime,[seq(i,i=3..N,2)]):
%p A339708 S:= sort(select(`<`,[seq(seq(P[i]*P[j],i=1..j),j=1..nops(P))],N)):
%p A339708 V:= Vector(N):
%p A339708 for p in P do
%p A339708 for s in S do
%p A339708   v:= p+s;
%p A339708     if v>N then break fi;
%p A339708   V[v]:= V[v]+1
%p A339708 od od:
%p A339708 seq(V[i],i=2..N,2);
%t A339708 {0}~Join~Array[Count[IntegerPartitions[2 #, {2}, All, -(# - 2)], _?(And[AnyTrue[#, PrimeQ], AnyTrue[#, PrimeOmega[#] == 2 &]] &)] &, 86, 2] (* _Michael De Vlieger_, Dec 13 2020 *)
%o A339708 (PARI) a(n) = {my(nb=0); forprime(p=3, 2*n, if (bigomega(2*n-p) == 2, nb++);); nb;} \\ _Michel Marcus_, Dec 14 2020
%Y A339708 Cf. A001358, A235645, A339709.
%K A339708 nonn,look
%O A339708 1,10
%A A339708 _J. M. Bergot_ and _Robert Israel_, Dec 13 2020