A224612 Let p = prime(n). Smallest j such that q = j*2*p^3-1, r = j*p*2*q^2-1, s = j*p*2*r^2-1, and j*p*2*s^2-1 are prime numbers.

29952, 12063, 1463, 6102, 11661, 49552, 639179, 2099290, 291248, 393186, 545251, 321303, 436641, 278295, 746832, 237852, 56490, 165901, 152847, 619755, 777177, 3410085, 117513, 2015421, 497170, 14750, 161190, 347039, 251835, 57536, 222, 2083286, 384944, 1228474, 3909960, 344164, 332224, 207751, 14060

Offset: 1

Author

Pierre CAMI, Apr 12 2013

Keywords

• nonn

Comments

Conjecture: a(n) exists for all n.

Crossrefs

Cf. A224492, A224609, A224610, A224611.

Programs

Maple

f:= proc(n) local j,p,q,r,s;
    p:= ithprime(n);
    for j from 1 do
      q:= j*2*p^3-1; if not isprime(q) then next fi;
      r:= j*p*2*q^2-1; if not isprime(r) then next fi;
      s:= j*p*2*r^2-1; if not isprime(s) then next fi;
      if isprime(j*p*2*s^2-1) then return j fi;
    od
end proc;
map(f, [$1..25]); # Robert Israel, May 15 2025

Mathematica

a[n_] := For[j = 1, j < 10^7, j++, p = Prime[n]; If[PrimeQ[q = j*2*p^3 - 1] && PrimeQ[r = j*2*p*q^2 - 1] && PrimeQ[s = j*2*p*r^2 - 1] && PrimeQ[j*2*p*s^2 - 1], Return[j]]]; Table[Print[an = a[n]]; an, {n, 1, 24}] (* Jean-François Alcover, Apr 12 2013 *)

Extensions

More terms from Jean-François Alcover, Apr 12 2013

Name clarified and more terms from Robert Israel, May 15 2025