A369250 Primes for which there is at least one representation as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

71, 103, 131, 151, 167, 191, 199, 211, 239, 251, 263, 271, 311, 331, 359, 383, 419, 431, 439, 467, 479, 487, 491, 503, 563, 587, 599, 607, 631, 647, 691, 719, 727, 739, 743, 751, 811, 823, 839, 859, 863, 887, 911, 919, 971, 983, 991, 1019, 1031, 1051, 1063, 1091, 1103, 1151, 1163, 1187, 1223, 1231, 1279, 1283, 1291

Offset: 1

Author

Antti Karttunen, Jan 22 2024

Keywords

• nonn

Comments

All such primes are by necessity of the form 4m+3 (in A002145). See A369249 for those 4m+3 primes that do not have such a representation.

Also by necessity, in these cases the primes in the sum (p*q + p*r + q*r) must all be distinct, that is, we actually need p < q < r, otherwise the sum would not be a prime.

Examples

			71 is present as 71 = (3*5) + (3*7) + (5*7) = A003415(105).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Antti Karttunen, Primes with such a representation vs. 4m+3 primes without such a representation (Plot2 comparison of the densities)

Crossrefs

Primes in A369251.

Setwise difference A002145 \ A369249.

Subsequence of A189441.

Cf. A003415, A369054, A369056.

Programs

PARI

isA369250(n) = (isprime(n) && (A369054(n)>0)); \\ Needs also program from A369054.