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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.

%I A224612 #15 May 15 2025 14:46:21
%S A224612 29952,12063,1463,6102,11661,49552,639179,2099290,291248,393186,
%T A224612 545251,321303,436641,278295,746832,237852,56490,165901,152847,619755,
%U A224612 777177,3410085,117513,2015421,497170,14750,161190,347039,251835,57536,222,2083286,384944,1228474,3909960,344164,332224,207751,14060
%N 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.
%C A224612 Conjecture: a(n) exists for all n.
%p A224612 f:= proc(n) local j,p,q,r,s;
%p A224612     p:= ithprime(n);
%p A224612     for j from 1 do
%p A224612       q:= j*2*p^3-1; if not isprime(q) then next fi;
%p A224612       r:= j*p*2*q^2-1; if not isprime(r) then next fi;
%p A224612       s:= j*p*2*r^2-1; if not isprime(s) then next fi;
%p A224612       if isprime(j*p*2*s^2-1) then return j fi;
%p A224612     od
%p A224612 end proc;
%p A224612 map(f, [$1..25]); # _Robert Israel_, May 15 2025
%t A224612 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 *)
%Y A224612 Cf. A224492, A224609, A224610, A224611.
%K A224612 nonn
%O A224612 1,1
%A A224612 _Pierre CAMI_, Apr 12 2013
%E A224612 More terms from _Jean-François Alcover_, Apr 12 2013
%E A224612 Name clarified and more terms from _Robert Israel_, May 15 2025