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A347436 a(n) is the least odd number that has exactly n decompositions as the sum of three primes, or 0 if there is no such odd number.

%I A347436 #8 Sep 03 2021 20:41:30
%S A347436 1,7,9,15,17,21,31,27,35,33,39,41,45,47,55,51,53,57,0,63,67,65,71,0,
%T A347436 79,81,0,85,77,83,99,0,0,89,97,95,103,111,101,0,0,0,115,107,0,129,121,
%U A347436 113,0,141,119,0,0,125,133,147,0,131,159,145,153,151,137,0,0,143,0,0,149,155,0,0,0,163,189
%N A347436 a(n) is the least odd number that has exactly n decompositions as the sum of three primes, or 0 if there is no such odd number.
%C A347436 Entries of 0 are conjectural.  If nonzero they are greater than 10^5.
%C A347436 Assuming Goldbach's conjecture, if k is odd then A068307(k) >= pi(k-4)-pi((k-1)/2). Using Pierre Dusart's bounds on pi(x), this implies that, for example, A068307(k) >= 4292 for odd k >= 10^5.  Thus (on the assumption of Goldbach's conjecture) the given entries of 0 are correct.
%e A347436 a(3) = 15 because 15 has exactly 3 decompositions as the sum of 3 primes: 2+2+11 = 3+5+7 = 5+5+5, and it is the smallest odd number that does.
%p A347436 N:= 10^5:
%p A347436 P:= select(isprime, [2,seq(i,i=3..N,2)]):
%p A347436 nP:=nops(P):
%p A347436 V:= Vector(N):
%p A347436 for i from 1 to nP do
%p A347436   for j from i to nP while P[i]+P[j] <= N do
%p A347436     for k from j to nP do
%p A347436       n:= P[i]+P[j]+P[k];
%p A347436       if n > N then break fi;
%p A347436       V[n]:= V[n]+1;
%p A347436 od od od:
%p A347436 R:= Vector(300):
%p A347436 for i from 1 to N by 2 do
%p A347436   if V[i] <= 300 and V[i] > 0 and R[V[i]] = 0 then R[V[i]]:= i fi
%p A347436 od:
%p A347436 convert(R,list);
%Y A347436 Cf. A068307, A139321.
%K A347436 nonn
%O A347436 0,2
%A A347436 _J. M. Bergot_ and _Robert Israel_, Sep 02 2021