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A261395 Primes p such that (prime(p)-1)^2 = (prime(q)-1)*(prime(r)-1) for some pair of distinct primes q and r.

%I A261395 #16 Aug 18 2015 11:16:27
%S A261395 13,47,137,191,193,223,227,313,701,857,907,947,991,1009,1069,1291,
%T A261395 1531,1889,2281,2411,2447,2647,3181,3389,3539,3593,4093,4099,4409,
%U A261395 4481,4603,4721,5557,5647,6581,6793,6869,6961,7211,7349,7523,7723,7753,8461,8537,8543,8807,9137,9241,9281
%N A261395 Primes p such that (prime(p)-1)^2 = (prime(q)-1)*(prime(r)-1) for some pair of distinct primes q and r.
%C A261395 Conjecture: Let d be any nonzero integer. Then there are infinitely many prime triples (p,q,r) with p,q,r distinct such that (prime(p)+d)^2 = (prime(q)+d)*(prime(r)+d). In other words, the set {prime(p)+d: p is prime} contains infinitely many nontrivial three-term geometric progressions.
%D A261395 Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
%H A261395 Zhi-Wei Sun, <a href="/A261395/b261395.txt">Table of n, a(n) for n = 1..600</a>
%H A261395 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
%e A261395 a(1) = 13 since (prime(13)-1)^2 = (41-1)^2 = 1600 = (5-1)*(401-1) = (prime(3)-1)*(prime(79)-1) with 13, 3, 79 all prime.
%e A261395 a(2) = 47 since (prime(47)-1)^2 = 210^2 = 44100 = 30*1470 = (prime(11)-1)*(prime(233)-1) with 47, 11, 233 all prime.
%t A261395 Dv[n_]:=Divisors[n]
%t A261395 L[n_]:=Length[Dv[n]]
%t A261395 f[n_]:=Prime[Prime[n]]-1
%t A261395 PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
%t A261395 n=0;Do[Do[If[PQ[Part[Dv[f[k]^2],i]+1]&&PQ[Part[Dv[f[k]^2],L[f[k]^2]-i+1]+1],n=n+1;Print[n," ",Prime[k]];Goto[aa]];Continue,{i,1,(L[f[k]^2]-1)/2}];
%t A261395 Label[aa];Continue,{k,1,1150}]
%Y A261395 Cf. A000040, A000290, A260082, A261352, A261385.
%K A261395 nonn
%O A261395 1,1
%A A261395 _Zhi-Wei Sun_, Aug 17 2015