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A377406 Primes which are sums of a prime triple (p, p+4, p+6).

%I A377406 #40 Nov 19 2024 10:19:29
%S A377406 31,211,1381,3271,4999,6421,8059,9769,10399,11551,16249,20479,23269,
%T A377406 23629,27031,28309,33349,35491,39019,54139,63949,70879,106591,109579,
%U A377406 116131,127219,130729,141439,142969,150151,151771,153589,163741,167449,169591,195511,205339,208489,210361,216679,222601,224149
%N A377406 Primes which are sums of a prime triple (p, p+4, p+6).
%C A377406 The prime triple herein is restricted to the form (p, p+4, p+6). In number theory, the other form of a prime triple is (p, p+2, p+6).
%C A377406 Equivalently, primes of the form 3*A022005(k) + 10.
%F A377406 a(n) mod 6 = 1.
%e A377406 The first term is 31, the sum of the triple (7, 11, 13).
%e A377406 The second term is 211, the sum of the triple (67, 71, 73).
%p A377406 q:= p-> andmap(isprime, (t->[p, t, t-4, t+2])((p+2)/3)):
%p A377406 select(q, [6*i+1$i=1..50000])[];  # _Alois P. Heinz_, Nov 13 2024
%t A377406 Select[Total /@ Select[Partition[Prime[Range[7500]], 3, 1], Differences[#] == {4, 2} &], PrimeQ] (* _Amiram Eldar_, Oct 31 2024 *)
%o A377406 (Python)
%o A377406 from sympy import isprime
%o A377406 sums = set()
%o A377406 for n in range(100000):
%o A377406     if isprime(n) and isprime(n+4) and isprime(n+6) and isprime(3*n+10):
%o A377406             sums.add(3*n+10)
%o A377406 print(sorted(sums))
%Y A377406 Cf. A022005, A162001, A376013.
%Y A377406 Subset of A002476.
%K A377406 nonn
%O A377406 1,1
%A A377406 _James S. DeArmon_, Oct 27 2024