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 A124867 Numbers that are the sum of 3 distinct primes. 11
 10, 12, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS (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. A125688(a(n)) > 0. - Reinhard Zumkeller, Nov 30 2006 LINKS Eric W. Weisstein, Goldbach conjecture Wikipedia, Goldbach's conjecture Wikipedia, Goldbach's weak conjecture EXAMPLE 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. 5 + 5 + 5 = 15, but note also 3 + 5 + 7 = 15, so 15 is in the sequence. Although 13 = 3 + 3 + 7 = 3 + 5 + 5, both of those repeat primes, so 13 is not in the sequence. MATHEMATICA threePrimes[n_] := Module[{p, q, r}, {p, q, r} /. Solve[n == p + q + r && p < q < r, {p, q, r}, Primes]]; 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 *) has3DistPrimesPart[n_] := Length[Select[IntegerPartitions[n, {3}], Length[Union[#]] == 3 && Union[PrimeQ[#]] == {True} &]] > 0; Select[Range[100], has3DistPrimesPart] (* Alonso del Arte, Apr 26 2020 *) PROG (PARI) a(n)=if(n>5, n+12, [10, 12, 14, 15, 16][n]) \\ Charles R Greathouse IV, Aug 26 2011 CROSSREFS Cf. A124868 (not the sum of 3 distinct primes), A068307, A125688. Sequence in context: A331276 A230597 A330904 * A199991 A161598 A122426 Adjacent sequences:  A124864 A124865 A124866 * A124868 A124869 A124870 KEYWORD nonn,easy AUTHOR Alexander Adamchuk, Nov 11 2006 STATUS approved

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Last modified April 14 00:42 EDT 2021. Contains 342941 sequences. (Running on oeis4.)