A224535 -id:A224535 - cp's OEIS frontend

A224534 Prime numbers that are the sum of three distinct prime numbers.

19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307

Offset: 1

T. D. Noe, Apr 15 2013

nonn

Similar to Goldbach's weak conjecture.
Primes in A124867, and by the comment in A124867 also the set of all primes >=19. - R. J. Mathar, Apr 19 2013
"Goldbach's original conjecture (sometimes called the 'ternary' Goldbach conjecture), written in a June 7, 1742 letter to Euler, states 'at least it seems that every number that is greater than 2 is the sum of three primes' (Goldbach 1742; Dickson 2005, p. 421). Note that here Goldbach considered the number 1 to be a prime, a convention that is no longer followed." [Weisstein] - Jonathan Vos Post, May 15 2013

			19 = 3 + 5 + 11.

H. A. Helfgott and David J. Platt, Numerical Verification of the Ternary Goldbach Conjecture up to 8.875e30, arXiv:1305.3062 [math.NT], 2013-2014.
H. A. Helfgott and David J. Platt, Numerical verification of the Ternary Goldbach Conjecture up to 8.875*10^30, Exp. Math. 22 (4) (2013) 406-409.
Eric W. Weisstein, Goldbach conjecture
Wikipedia, Goldbach's conjecture
Wikipedia, Goldbach's weak conjecture

Cf. A002372, A002375, A024684 (number of sums), A224535, A166063, A166061, A071621.

Union[Select[Total /@ Subsets[Prime[Range[2, 30]], {3}], PrimeQ]]

A347933 Positive integers that can't be written in the form a+b+c for some positive integers a, b, and c satisfying gcd(a,b)=1, gcd(a,c)>1, and gcd(b,c)>1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 24, 30, 36, 42, 48, 60, 84, 90, 210

Offset: 1

José Hernández, Sep 20 2021

			1 and 2 are the first two terms of the sequence because they can't even be written as sums of three positive integers.
3 is the third term of the sequence because there is only one way to express it as a sum of three positive integers (1+1+1).
11 does not belong to the sequence because 11=2+3+6 (and gcd(2,3)=1, gcd(2,6)=2, and gcd(3,6)=3).

Freddy Barrera, Bernardo Recamán, and Stan Wagon, Sums of triples with one pair relatively prime. American Mathematical Monthly, 127 (2020), no. 1, pp. 89-90.

Cf. A224535, A100951, A100952.

Select[Range@210,Select[Flatten[Permutations/@IntegerPartitions[#,{3}],1],GCD[#[[1]],#[[2]]]==1&&GCD[#[[1]],#[[3]]]>1&&GCD[#[[2]],#[[3]]]>1&]=={}&] (* Giorgos Kalogeropoulos, Sep 20 2021 *)

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A224534 Prime numbers that are the sum of three distinct prime numbers.

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