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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.

Cf. A068307, A139321.

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allocated for Robert Israela(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.

1, 7, 9, 15, 17, 21, 31, 27, 35, 33, 39, 41, 45, 47, 55, 51, 53, 57, 0, 63, 67, 65, 71, 0, 79, 81, 0, 85, 77, 83, 99, 0, 0, 89, 97, 95, 103, 111, 101, 0, 0, 0, 115, 107, 0, 129, 121, 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

0,2

Entries of 0 are conjectural. If nonzero they are greater than 10^5.

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.

N:= 10^5:

P:= select(isprime, [2, seq(i, i=3..N, 2)]):

nP:=nops(P):

V:= Vector(N):

for i from 1 to nP do

for j from i to nP while P[i]+P[j] <= N do

for k from j to nP do

n:= P[i]+P[j]+P[k];

if n > N then break fi;

V[n]:= V[n]+1;

od od od:

R:= Vector(300):

for i from 1 to N by 2 do

if V[i] <= 300 and V[i] > 0 and R[V[i]] = 0 then R[V[i]]:= i fi

od:

convert(R, list);

Cf. A139321.

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J. M. Bergot and Robert Israel, Sep 02 2021

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allocated for Robert Israel

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