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A339709 a(n) is the least even number that has exactly n decompositions as the sum of an odd prime and a semiprime, or 0 if there is no such number.

%I A339709 #9 Jan 14 2021 18:14:47
%S A339709 2,12,20,26,28,44,38,52,76,80,62,92
%N A339709 a(n) is the least even number that has exactly n decompositions as the sum of an odd prime and a semiprime, or 0 if there is no such number.
%C A339709 a(n) is the least even number 2*k such that A339708(k)=n.
%C A339709 Conjecture: a(12) = 0, but a(n) > 0 for all other n.
%C A339709 Since this is only a conjecture, the Data stops at a(11)=92.
%C A339709 For n>=13, the sequence continues with 98, 136, 128, 122, 158, 166, 152, 206, 188, 222, 242, 232, 248, 266, 272, 296, 320, 308, 352, 382, 412, 326, 418, 402, 440, 454, 398, 492, 458, 488, 500, 554, 542, 518, 578, 618, 572, 626, 642, 678, 632, ...
%H A339709 Robert Israel, <a href="/A339709/a339709.txt">Table of n, a(n) for n = 0 .. 800</a> (with conjectured a(12)=0).
%e A339709 a(4) = 28 because we can write 28 = 3+25 = 7+21 = 13+15 = 19+9 where 3, 7, 13 and 19 are odd primes and 25, 21, 15, and 9 are semiprimes, and 28 is the least even number with exactly 4 decompositions of this type.
%p A339709 N:= 10000:
%p A339709 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):
%p A339709 for p in P do
%p A339709   for s in S while p+s<=N do V[p+s]:= V[p+s]+1 od
%p A339709 od:R:= Array(0..max(V)):for i from 2 to N by 2 do
%p A339709   v:= V[i];
%p A339709   if R[v] = 0 then R[v]:= i fi
%p A339709 od:convert(R[1..500],list);
%Y A339709 Cf. A339708.
%K A339709 nonn,more
%O A339709 0,1
%A A339709 _J. M. Bergot_ and _Robert Israel_, Dec 13 2020