A358155 First of four consecutive primes p,q,r,s such that (2*p+q)/5, (q+r)/10 and (r+2*s)/5 are prime.

11, 2696717, 3500381, 3989903, 4515113, 8164073, 12451013, 18793013, 23567267, 24057713, 30312409, 36391853, 44569853, 45657881, 53442343, 54721253, 54944761, 56652203, 63993803, 76763081, 90662303, 92889127, 94670143, 105790973, 106339481, 108988223, 117213871, 118802533, 130741007, 145543523

Offset: 1

Author

J. M. Bergot and Robert Israel, Nov 01 2022

Keywords

• nonn

Comments

11 is the only term that is in A007530, because if p is in A007530 (so q = p+2, r = p+6 and s = p+8), one of p, q, r, s, 2*p+q, q+r and r+2*s is divisible by 7.

Examples

			a(3) = 3500381 is a term because 3500381, 3500383, 3500407, 3500429 are four consecutive primes with (2*3500381 + 3500383)/5 = 2100229, (3500383 + 3500407)/10 = 700079, and (3500407 + 2*3500429)/5 = 2100253 all prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..250

Crossrefs

Cf. A007530, A358149.

Programs

Maple

Res:= NULL: count:= 0:
q:= 2: r:= 3: s:= 5:
while count < 40 do
  p:= q; q:= r; r:= s; s:= nextprime(s);
  t:= (2*p+q)/5; u:= (q+r)/10; v:= (r+2*s)/5;
  if (t::integer and u::integer and v::integer and isprime(t) and isprime(u) and isprime(v)) then
    count:= count+1; Res:= Res, p;
  fi
od:
Res;

Mathematica

Select[Partition[Prime[Range[8.3*10^6]], 4, 1], PrimeQ[(2*#[[1]] + #[[2]])/5] && PrimeQ[(#[[2]] + #[[3]])/10] && PrimeQ[(#[[3]] + 2*#[[4]])/5] &][[;; , 1]] (* Amiram Eldar, Nov 01 2022 *)