A124867 Numbers that are the sum of 3 distinct primes.

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

Offset: 1

Author

Alexander Adamchuk, Nov 11 2006

Keywords

• nonn
• easy

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

Examples

			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.
		

Links

• Eric W. Weisstein, Goldbach conjecture
• Wikipedia, Goldbach's conjecture
• Wikipedia, Goldbach's weak conjecture

Crossrefs

Cf. A124868 (not the sum of 3 distinct primes), A068307, A125688.

Programs

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 *)
Union[Total/@Subsets[Prime[Range[20]],{3}]] (* Harvey P. Dale, Feb 06 2024 *)

PARI

a(n)=if(n>5,n+12,[10, 12, 14, 15, 16][n]) \\ Charles R Greathouse IV, Aug 26 2011