A288632 -id:A288632 - cp's OEIS frontend

A339775 Primes p such that (p+nextprime(p))/6 is prime and 6*p is the sum of two consecutive primes.

5, 7, 13, 37, 127, 389, 719, 937, 3089, 7669, 9199, 12211, 17099, 17519, 18919, 19259, 19273, 19853, 20063, 21379, 22453, 22643, 23059, 23143, 23173, 23753, 24113, 24329, 25339, 25873, 31387, 31667, 32803, 33203, 34057, 34183, 36629, 37253, 37831, 37967, 38557, 39293, 40429, 41039, 42743, 48163

Offset: 1

J. M. Bergot and Robert Israel, Dec 16 2020

nonn

			a(3) = 13 is a term because 13 is prime, the next prime is 17, (13+17)/6 = 5 is prime, and 6*13 = 78 = 37 + 41 is the sum of consecutive primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A163487 and A288632.

filter:= proc(p) local q;
  if not isprime(p) then return false fi;
  q:= prevprime(3*p);
  if q + nextprime(q) <> 6*p then return false fi;
  q:= (p+nextprime(p))/6;
  q::integer and isprime(q)
end proc:
select(filter, [seq(i,i=3..10^5,2)]);

A351373 a(n) is the least prime that begins a sequence of 2*n consecutive primes whose sum is 6 times a prime.

Original entry on oeis.org

5, 19, 89, 43, 103, 7, 19, 3, 19, 5, 67, 19, 31, 151, 19, 3, 3, 5, 61, 127, 61, 103, 13, 13, 137, 109, 149, 67, 59, 103, 59, 271, 983, 31, 3, 43, 277, 181, 3, 683, 307, 307, 83, 313, 181, 193, 331, 191, 151, 157, 151, 127, 151, 421, 523, 97, 97, 3, 5, 61, 61, 61, 331, 283, 283, 61, 167, 2003, 263

Offset: 1

J. M. Bergot and Robert Israel, Feb 09 2022

nonn

Conjecture: each odd prime occurs infinitely many times.

			a(3) = 89 because the sum of the 6 consecutive primes starting with 89 is 89+97+101+103+107+109 = 606 = 6*101 where 101 is prime, and no smaller prime works.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A288632.

P:= select(isprime,[seq(i,i=3..10^5,2)]):
S:= [0,op(ListTools:-PartialSums(P))]:
nP:= nops(S):
f:= proc(n) local i, s;
  for i from 1 to nP-2*n do
    s:= S[i+2*n]-S[i];
    if s mod 6 = 0 and isprime(s/6) then return P[i] fi
  od
end proc:
map(f, [$1..100]);

cp's OEIS Frontend

A339775 Primes p such that (p+nextprime(p))/6 is prime and 6*p is the sum of two consecutive primes.

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