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%I #9 Jan 14 2021 18:14:47
%S 2,12,20,26,28,44,38,52,76,80,62,92
%N 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 a(n) is the least even number 2*k such that A339708(k)=n.
%C Conjecture: a(12) = 0, but a(n) > 0 for all other n.
%C Since this is only a conjecture, the Data stops at a(11)=92.
%C 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 Robert Israel, <a href="/A339709/a339709.txt">Table of n, a(n) for n = 0 .. 800</a> (with conjectured a(12)=0).
%e 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 N:= 10000:
%p 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 for p in P do
%p for s in S while p+s<=N do V[p+s]:= V[p+s]+1 od
%p od:R:= Array(0..max(V)):for i from 2 to N by 2 do
%p v:= V[i];
%p if R[v] = 0 then R[v]:= i fi
%p od:convert(R[1..500],list);
%Y Cf. A339708.
%K nonn,more
%O 0,1
%A _J. M. Bergot_ and _Robert Israel_, Dec 13 2020
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