A340463 Primes p such that p*q+r*s+t*u is prime, where p,q,r,s,t,u are consecutive primes.

3, 17, 41, 47, 67, 107, 193, 197, 199, 211, 229, 239, 313, 331, 367, 461, 467, 503, 523, 571, 919, 929, 991, 1021, 1039, 1093, 1109, 1163, 1193, 1237, 1277, 1327, 1361, 1381, 1621, 1627, 1783, 1901, 2029, 2099, 2143, 2381, 2389, 2423, 2473, 2663, 2677, 2801, 2917, 2939, 2953, 2957, 2963, 3019

Offset: 1

Author

J. M. Bergot and Robert Israel, Jan 08 2021

Keywords

• nonn

Examples

			a(3)=41 is a term because 41*43+47*53+59*61=7853 is prime, where 41,43,47,53,59,61 are consecutive primes.
		

Links

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

Crossrefs

Cf. A340464.

Programs

Maple

map(ithprime, select(i -> isprime(ithprime(i)*ithprime(i+1)+ithprime(i+2)*ithprime(i+3)+ithprime(i+4)*ithprime(i+5)), [$1..1000]));

Mathematica

Select[Partition[Prime[Range[500]],6,1],PrimeQ[#[[1]]#[[2]]+#[[3]]#[[4]]+#[[5]]#[[6]]]&][[;;,1]] (* Harvey P. Dale, Jan 10 2025 *)

Python

from sympy import nextprime, isprime
def aupto(nn):
  alst, consec6 = [], [2, 3, 5, 7, 11, 13]
  p, q, r, s, t, u = consec6; prod = p*q+r*s+t*u
  while p <= nn:
    if isprime(prod): alst.append(p)
    consec6 = consec6[1:] + [nextprime(consec6[-1])]
    p, q, r, s, t, u = consec6; prod = p*q+r*s+t*u
  return alst
print(aupto(3019)) # Michael S. Branicky, Jan 08 2021