A152012 -id:A152012 - cp's OEIS frontend

A178763 Product of primitive prime factors of Fibonacci(n).

1, 1, 2, 3, 5, 1, 13, 7, 17, 11, 89, 1, 233, 29, 61, 47, 1597, 19, 4181, 41, 421, 199, 28657, 23, 3001, 521, 5777, 281, 514229, 31, 1346269, 2207, 19801, 3571, 141961, 107, 24157817, 9349, 135721, 2161, 165580141, 211, 433494437, 13201, 109441, 64079

Offset: 1

T. D. Noe, Jun 10 2010

nonn

Same as A001578 for the first 18 terms.

Let b(n) be the greatest divisor of Fibonacci(n) that is coprime to Fibonacci(m) for all positive integers m < n, then a(n) = b(n) for all n, provided that no Wall-Sun-Sun prime exists. Otherwise, if p is a Wall-Sun-Sun prime and A001177(p) = k (then A001177(p^2) = k), then p^2 divides b(k), but by definition a(k) is squarefree. - Jianing Song, Jul 02 2019

T. D. Noe, Table of n, a(n) for n = 1..1000
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See pp. 60-64 for a table of the first 385 terms.
Florian Luca, Carl Pomerance, and Stephen Wagner, Fibonacci Integers (preprint)

Cf. A061446, A086597, A152012 (Indices of prime terms).

a(n)=my(d=divisors(n), f=fibonacci(n), t); t=lcm(apply(fibonacci,d[1..#d-1])); while((t=gcd(t,f))>1, f/=t); f \\ Charles R Greathouse IV, Nov 30 2016

a(n) = A061446(n) / A178764(n).

a(n) = A061446(n) / gcd(A061446(n), n) if n != 5, 6, provided that no Wall-Sun-Sun prime exists. - Jianing Song, Jul 02 2019

A178762 Prime numbers that are Fibonacci integers.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 61, 89, 107, 199, 211, 233, 281, 421, 521, 1103, 1597, 2161, 2207, 2521, 3001, 3571, 5779, 9349, 9901, 14503, 19801, 28657, 90481, 103681, 109441, 135721, 141961, 514229, 3010349, 6376021, 11128427

Offset: 1

T. D. Noe, Jun 10 2010

nonn

A Fibonacci integer is a number that can be written as the product and/or quotient of Fibonacci numbers (A000045). For example, 107 is a Fibonacci integer because Fib(36)/(Fib(18)*Fib(3)*Fib(4)^3) = 107. Observe that the prime Fibonacci numbers (A005478) are a subset of these primes. Luca, Pomerance, and Wagner conjecture that this sequence is infinite. The paper's Remark 2 and sequences A152012, A178763, and A178764 are useful in finding these primes.

Florian Luca, Carl Pomerance, and Stephan Wagner, Fibonacci Integers (preprint)

A152013 Indices of Fibonacci numbers having at least two distinct primitive prime factors.

Original entry on oeis.org

19, 27, 31, 37, 41, 44, 46, 49, 50, 53, 55, 57, 58, 59, 61, 64, 67, 68, 69, 70, 71, 73, 77, 78, 79, 80, 81, 84, 85, 86, 87, 88, 89, 90, 91, 92, 95, 96, 97, 99, 100, 101, 102, 103, 104, 107, 109, 110, 113, 114, 115, 116, 117, 118, 120, 126, 127, 128, 129, 130, 133, 134

Offset: 1

Max Alekseyev, Nov 19 2008

nonn

It is known that Fibonacci number A000045(n) has a primitive prime factor for all n, except n=0, 1, 2, 6 and 12. This sequence lists such indices n that A000045(n) has at least two primitive prime factors. Its sister sequence A152012 provides indices of Fibonacci numbers with exactly one primitive prime factor. The current sequence A152013 and its sister sequence A152012 along with the finite set {0,1,2,6,12} form a partition of the natural numbers.

Cf. A000045, A001578, A152012.

A262341 Largest primitive prime factor of Fibonacci number F(n), or 1 if no primitive.

Original entry on oeis.org

1, 1, 2, 3, 5, 1, 13, 7, 17, 11, 89, 1, 233, 29, 61, 47, 1597, 19, 113, 41, 421, 199, 28657, 23, 3001, 521, 109, 281, 514229, 31, 2417, 2207, 19801, 3571, 141961, 107, 2221, 9349, 135721, 2161, 59369, 211, 433494437, 307, 109441, 461, 2971215073, 1103, 6168709, 151

Offset: 1

Jonathan Sondow, Oct 12 2015

nonn

Carmichael proved that a(n) > 1 if n > 12.

See A001578 (smallest primitive prime factor of F(n)) and A061446 (primitive part of F(n)) for more links.

			The prime factors of F(46)= 139 * 461 * 28657 that do not divide any smaller Fibonacci number are 139 and 461, so a(46) = 461.

R. D. Carmichael, On the numerical factors of the arithmetic forms α^n±β^n, Annals of Math., 15 (1913), 30-70.
Blair Kelly, Fibonacci and Lucas Factorizations
Ron Knott, Fibonacci numbers and special prime factors
Wikipedia, Carmichael's theorem

Cf. A000045, A001578, A061446, A086597, A152012, A152013.

prms={}; Table[f=First/@FactorInteger[Fibonacci[n]]; p=Complement[f, prms]; prms=Join[prms, p]; If[p=={}, 1, Last[p]], {n, 50}]

use ntheory ":all"; my %s; for (1..100) { my @f = factor(lucasu(1,-1,$)); pop @f while @f && $s{$f[-1]}++; say "$ ", $f[-1] || 1; }  # Dana Jacobsen, Oct 13 2015

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

Original entry on oeis.org

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

Eric Chen, Nov 29 2014

nonn

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.

			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.

Cf. A152012 (for Fibonacci numbers).

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

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 *)

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

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

More terms from Colin Barker, Nov 30 2014

cp's OEIS Frontend

A178763 Product of primitive prime factors of Fibonacci(n).

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A178762 Prime numbers that are Fibonacci integers.

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A152013 Indices of Fibonacci numbers having at least two distinct primitive prime factors.

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A262341 Largest primitive prime factor of Fibonacci number F(n), or 1 if no primitive.

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

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