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Smallest k such that k*2*p(n)^2-1=q is prime, k*2*q^2-1=r, k*2*r^2-1=s, k*2*r^2-1=t, r, s, and t are also prime.
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%I #23 Apr 12 2013 08:25:55

%S 5103,36189,7315,29608,128115,3496,64590,143079,83919,5586,13209,2833,

%T 235339,61621,164349,2668,84574,1140,47335,108079,7978,181366,146140,

%U 2616,165864,86100,11455,8925,23191,197938,28194,229309,196236,274186

%N Smallest k such that k*2*p(n)^2-1=q is prime, k*2*q^2-1=r, k*2*r^2-1=s, k*2*r^2-1=t, r, s, and t are also prime.

%C conjecture: a(n) exist for all n

%C t=k*2*(k*2*(k*2*(k*2*p(n)^2-1)^2-1)^2-1)^2-1

%C s=k*2*(k*2*(k*2*p(n)^2-1)^2-1)^2-1

%C r=k*2*(k*2*p(n)^2-1)^2-1

%C q=k*2*p(n)^2-1

%H Pierre CAMI, <a href="/A224492/b224492.txt">Table of n, a(n) for n = 1..80</a>

%t a[n_] := For[k = 1, True, k++, p = Prime[n]; If[PrimeQ[q = k*2*p^2 - 1] && PrimeQ[r = k*2*q^2 - 1] && PrimeQ[s = k*2*r^2 - 1] && PrimeQ[k*2*s^2 - 1], Return[k]]]; Table[Print[an = a[n]]; an, {n, 1, 34}] (* _Jean-François Alcover_, Apr 12 2013 *)

%o (PFGW & SCRIPTIFY)

%o SCRIPT

%o DIM k

%o DIM i,0

%o DIM q

%o DIMS t

%o OPENFILEOUT myf,a(n).txt

%o LABEL a

%o SET i,i+1

%o IF i>34 THEN END

%o SET k,0

%o LABEL b

%o SET k,k+1

%o SETS t,%d,%d,%d\,;k;i;p(i)

%o SET q,k*2*p(i)^2-1

%o PRP q,t

%o IF ISPRP THEN GOTO c

%o GOTO b

%o LABEL c

%o SET q,k*2*q^2-1

%o PRP q,t

%o IF ISPRP THEN GOTO d

%o GOTO b

%o LABEL d

%o SET q,k*2*q^2-1

%o PRP q,t

%o IF ISPRP THEN GOTO e

%o GOTO b

%o LABEL e

%o SET q,k*2*q^2-1

%o PRP q,t

%o IF ISPRP THEN WRITE myf,t

%o IF ISPRP THEN GOTO a

%o GOTO b

%Y Cf. A224489, A224490, A224491.

%K nonn

%O 1,1

%A _Pierre CAMI_, Apr 08 2013

%E Typo in name fixed by _Zak Seidov_, Apr 11 2013