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

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, 42, 50, 53, 54, 56, 58, 59, 66, 69, 89, 90, 94, 95, 97, 99, 101, 104, 117, 118, 120, 135, 138, 160, 167, 181, 191, 210, 221, 237, 242, 247

Offset: 1

Author

Eric Chen, Nov 29 2014

Keywords

• nonn

Comments

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.)

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.

Examples

			Pell(1) = 1, which has no prime factors, so 1 is not in this sequence.
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.
Pell(5) = 29, which is a prime and the only primitive prime factor of itself, so 5 is in this sequence.
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.
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.
Pell(19) = 6625109 = 37 * 179057, both of which are primitive prime factors of Pell(19), so 19 is not in this sequence.
		

Crossrefs

Cf. A152012 (for Fibonacci numbers).

Cf. A000129, A246556, A096650, A086383, A008555, A175181.

Programs

Mathematica

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
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 *)

PARI

pell(n) = imag((1 + quadgen(8))^n);
isok(pf, vp) = sum(i=1, #pf, vecsearch(vp, pf[i]) == 0) == 1;
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

Extensions

Two incorrect terms (72 and 110) deleted by Colin Barker, Nov 29 2014

More terms from Colin Barker, Nov 30 2014