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A124867 Numbers that are the sum of 3 distinct primes.

%I A124867 #34 Feb 16 2025 08:33:03
%S A124867 10,12,14,15,16,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,
%T A124867 36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,
%U A124867 59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81
%N A124867 Numbers that are the sum of 3 distinct primes.
%C A124867 (Conjecture) Every number n > 17 is the sum of 3 distinct primes. Natural numbers that are not the sum of 3 distinct primes are listed in A124868.
%C A124867 A125688(a(n)) > 0. - _Reinhard Zumkeller_, Nov 30 2006
%H A124867 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/GoldbachConjecture.html">Goldbach conjecture</a>
%H A124867 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach&#39;s_conjecture">Goldbach's conjecture</a>
%H A124867 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach&#39;s_weak_conjecture">Goldbach's weak conjecture</a>
%e A124867 The first three primes are 2, 3, 5, and 2 + 3 + 5 = 10, so 10 is in the sequence. No smaller integer is in the sequence.
%e A124867 5 + 5 + 5 = 15, but note also 3 + 5 + 7 = 15, so 15 is in the sequence.
%e A124867 Although 13 = 3 + 3 + 7 = 3 + 5 + 5, both of those repeat primes, so 13 is not in the sequence.
%t A124867 threePrimes[n_] := Module[{p, q, r}, {p, q, r} /. Solve[n == p + q + r && p < q < r, {p, q, r}, Primes]];
%t A124867 Reap[For[n = 10, n <= 100, n++, sol = threePrimes[n]; If[MatchQ[sol, {{_, _, _}..}], Print[n, " ", sol[[1]]]; Sow[n]]]][[2, 1]] (* _Jean-François Alcover_, Apr 26 2020 *)
%t A124867 has3DistPrimesPart[n_] := Length[Select[IntegerPartitions[n, {3}], Length[Union[#]] == 3 && Union[PrimeQ[#]] == {True} &]] > 0; Select[Range[100], has3DistPrimesPart] (* _Alonso del Arte_, Apr 26 2020 *)
%t A124867 Union[Total/@Subsets[Prime[Range[20]],{3}]] (* _Harvey P. Dale_, Feb 06 2024 *)
%o A124867 (PARI) a(n)=if(n>5,n+12,[10, 12, 14, 15, 16][n]) \\ _Charles R Greathouse IV_, Aug 26 2011
%Y A124867 Cf. A124868 (not the sum of 3 distinct primes), A068307, A125688.
%K A124867 nonn,easy
%O A124867 1,1
%A A124867 _Alexander Adamchuk_, Nov 11 2006