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A247250 Indices of Pell numbers having exactly one primitive prime factor.

%I A247250 #57 Jun 08 2015 02:27:57
%S A247250 2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,18,21,24,29,30,32,33,35,38,41,
%T A247250 42,50,53,54,56,58,59,66,69,89,90,94,95,97,99,101,104,117,118,120,135,
%U A247250 138,160,167,181,191,210,221,237,242,247
%N A247250 Indices of Pell numbers having exactly one primitive prime factor.
%C A247250 Conjecture: The n-th Pell number A000129(n) has a primitive prime factor for all n > 1. (The n-th Fibonacci number A000045(n) has a primitive prime factor for all n except n = 0, 1, 2, 6, and 12.)
%C A247250 For prime p, all prime factors of Pell(p) are primitive. Hence the only primes in this sequence are the prime numbers in A096650, which gives the indices of prime Pell numbers.
%e A247250 Pell(1) = 1, which has no prime factors, so 1 is not in this sequence.
%e A247250 Pell(4) = 12 = 2^2 * 3, but 2 is not a primitive prime factor, and 3 is the only primitive prime factor of Pell(4), so 4 is in this sequence.
%e A247250 Pell(5) = 29, which is a prime and the only primitive prime factor of itself, so 5 is in this sequence.
%e A247250 Pell(12) = 13860 = 2^2 * 3^2 * 5 * 7 * 11, but none of 2, 3, 5, 7 is a primitive prime factor, and 11 is the only primitive prime factor of Pell(12), so 12 is in this sequence.
%e A247250 Pell(14) = 80782 = 2 * 13^2 * 239, but neither 2 nor 13 is a primitive prime factor, and 239 is the only primitive prime factor of Pell(14), so 14 is in this sequence.
%e A247250 Pell(19) = 6625109 = 37 * 179057, both of which are primitive prime factors of Pell(19), so 19 is not in this sequence.
%t A247250 Select[Range[1000], PrimePowerQ[(1-Sqrt[2])^EulerPhi[#]*Cyclotomic[#, (1+Sqrt[2])/(1-Sqrt[2])]/GCD[Cyclotomic[#, (1+Sqrt[2])/(1-Sqrt[2])], # ]]&] - _Eric Chen_, Dec 12 2014
%t A247250 pell[n_] := pell[n] = ((1+Sqrt[2])^n-(1-Sqrt[2])^n )/(2*Sqrt[2]) // Round; primitivePrimeFactors[n_] := Cases[FactorInteger[pell[n]][[All, 1]], p_ /; And @@ (GCD[p, #] == 1 & /@ Array[pell, n-1])]; Reap[For[n=2, n <= 200, n++, If[Length[primitivePrimeFactors[n]] == 1, Print[n]; Sow[n]]]][[2, 1]] (* _Jean-François Alcover_, Dec 12 2014 *)
%o A247250 (PARI) pell(n) = imag((1 + quadgen(8))^n);
%o A247250 isok(pf, vp) = sum(i=1, #pf, vecsearch(vp, pf[i]) == 0) == 1;
%o A247250 lista(nn) = {vp = []; for (n=2, nn, pf = factor(pell(n))[,1]; if (isok(pf, vp), print1(n, ", ")); vp = vecsort(concat(vp, pf),, 8););} \\ _Michel Marcus_, Nov 29 2014
%Y A247250 Cf. A152012 (for Fibonacci numbers).
%Y A247250 Cf. A000129, A246556, A096650, A086383, A008555, A175181.
%K A247250 nonn
%O A247250 1,1
%A A247250 _Eric Chen_, Nov 29 2014
%E A247250 Two incorrect terms (72 and 110) deleted by _Colin Barker_, Nov 29 2014
%E A247250 More terms from _Colin Barker_, Nov 30 2014