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%I #4 Mar 30 2012 18:39:59
%S 19,23,29,31,37,41,0,0,47,0,53,0,59,0,67,0,73,71,79,0,0,0,0,0,83,89,0,
%T 97,0,0,103,101,0,0,0,0,107,0,113,0,0,0,0,127,0,0,0,139,0,0,131,0,0,
%U 137,151,0,0,0,0,149,0,157,0,163,0,0,0,0,0,0,0,0,167,181,0,0,0,0,179,173,0
%N Smallest prime which is equal to the sum of three distinct primes in exactly n ways, or 0 if no such prime exists.
%e a(1)=19 since 19=3+5+11; a(2)=23 since 23=5+7+11=3+7+13.
%e a(5)=37 since 37=3+5+29=3+11+23=5+13+19=7+11+19=7+13+17.
%t nn=100; t=Table[0, {nn}]; s=Table[{}, {nn}]; Do[p=Prime[i]+Prime[j]+Prime[k]; If[PrimeQ[p], pin=PrimePi[p]; If[pin<=nn, t[[pin]]++; AppendTo[s[[pin]], {i, j, k}]]], {i, nn}, {j, i+1, nn}, {k, j+1, nn}]; Table[pos=Position[t, n]; If[pos=={}, 0, Prime[pos[[1, 1]]]], {n, nn}]
%K nonn
%O 1,1
%A _Giovanni Teofilatto_, Nov 23 2004
%E Corrected and extended by _T. D. Noe_, Nov 23 2004
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