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A124867
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Numbers that are the sum of 3 distinct primes.
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11
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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
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OFFSET
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1,1
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COMMENTS
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(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
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LINKS
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Table of n, a(n) for n=1..69.
Eric W. Weisstein, Goldbach conjecture
Wikipedia, Goldbach's conjecture
Wikipedia, Goldbach's weak conjecture
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n)=if(n>5, n+12, [10, 12, 14, 15, 16][n]) \\ Charles R Greathouse IV, Aug 26 2011
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CROSSREFS
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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
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KEYWORD
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nonn,easy
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AUTHOR
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Alexander Adamchuk, Nov 11 2006
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STATUS
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approved
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