A001578 -id:A001578 - cp's OEIS frontend

A172115 Partial sums of A001605.

3, 7, 12, 19, 30, 43, 60, 83, 112, 155, 202, 285, 416, 553, 912, 1343, 1776, 2225, 2734, 3303, 3874, 6845, 11568, 16955, 26266, 35943, 50374, 75935, 106692, 142691, 180202, 231035, 312874, 417785, 547806, 695897, 897004, 1294383, 1728164, 2318205, 2911894, 3516605

Offset: 1

Jonathan Vos Post, Jan 25 2010

			a(1) = 3.
a(2) = 3 + 4 = 7.
a(3) = 3 + 4 + 5 = 12.

Amiram Eldar, Table of n, a(n) for n = 1..56 (calculated from the data at A001605)

Cf. A001605, A001578, A005478, A086597, A080345.

a(40) onwards from Amiram Eldar, Jul 22 2025

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

A285314 Numbers k such that the k-th term of some (generalized) Lucas sequence has no primitive prime factor.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 18, 30

Offset: 1

Tomohiro Yamada, Apr 17 2017

For a generalized Lucas sequence {U(n)} = {(a^n - b^n)/(a - b)}, where a + b and ab are nonzero coprime integers and (a/b) is not a root of unity, a prime factor of the n-th term of some Lucas sequence U(n) is called primitive if it does not divide U(r) for any r < n.
Let P = a + b > 0, Q = ab and D = P^2 - 4Q = (a - b)^2. In the case a, b are real (equivalently, D > 0), Carmichael shows that, if n <> 1, 2, 6, then U(n) has at least one primitive prime factor not dividing D except U(3) ((P, Q, D) = (1, -2, 9), (1, -1, 5)), U(5) ((P, Q, D) = (1, -1, 5)) and U(12) ((P, Q, D) = (1, -1, 5)).
Voutier determines the cases U(n) has no primitive prime factor for n = 5, 7 <= n <= 30. For n = 5, 7 <= n <= 30, any prime factor of U(n) divides D or U_r for some r < n in the following cases:
U(5): (P, Q, D) = (1, -1, 5)*, (1, 2, -7), (1, 3, -11), (1, 4, -15)*, (2, 11, -40)*, (12, 55, -76), (12, 377, -1364),
U(7): (P, Q, D) = (1, 2, -7)*, (1, 5, -19),
U(8): (P, Q, D) = (1, 2, -7), (2, 7, -24),
U(10): (P, Q, D) = (2, 3, -8), (5, 7, -3), (5, 18, -47),
U(12): (P, Q, D) = (1, -1, 5), (1, 2, -7), (1, 3, -11), (1, 4, -15), (1, 5, -19), (2, 15, -56),
U(13), U(18), U(30): (P, Q, D) = (1, 2, -7).
(The symbol * indicates that any prime factor of the corresponding U(n) divides D)
Bilu, Hanrot and Voutier shows that these are all for n = 5, n >= 7. Hence this sequence is complete.
From Jianing Song, Feb 23 2019: (Start)
Let {U(n)} be a generalized Lucas sequence. If p is a prime U(p) has no primitive prime factor, then U(p) = +-1. In contrast, if k is a composite number such that U(k) has no primitive prime factor, then U(k) cannot be +-1. As a result, the possible solutions to U(k) = +-1 for some Lucas sequence are k = 1, 2, 3, 5, 7, 13.
From U(1) = 1, U(2) = P, U(3) = P^2 - Q, U(4) = P*(P^2 - 2*Q), U(6) = P*(P^2 - Q)*(P^2 - 3*Q) we can see that for k = 1, 2, 3, 4, 6, there are infinitely many Lucas sequences such that U(k) has no primitive prime factor. Also, for p = 2, 3, there are infinitely many Lucas sequences such that any prime factor of U(p) divides D.
This sequence lists also numbers k such that E(p) = k has no solution over the primes for some Lucas sequence {U(n)}, where E(p) is the entry point of {U(n)} modulo p, that is, the smallest m > 0 such that p divides U(m). (End)

			If (P, Q, D) = (1, -1, 5) (giving the Fibonacci sequence), U(12) = 144 = 2^4 * 3^2, while U(4) = 3 and U(6) = 8 = 2^3. Hence U(12) with (P, Q, D) = (1, -1, 5) has no primitive prime factor and 12 belongs to this sequence.

Y. Bilu, G. Hanrot, P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers J. reine Angew. Math. 539 (2001), 75--122, a preprint version available from here.
R. D. Carmichael, On the numerical factors of the arithmetic forms a^n +- b^n, Ann. of Math., 15 (1913), 30--70.
P. M. Voutier, Primitive divisors of Lucas and Lehmer sequences, Math. Comp. 64 (1995), 869--888.
M. Yabuta, A simple proof of Carmichael's theorem on primitive divisors, Fibonacci Quart., 39 (2001), 439--443.

Cf. A001578, A058036, A246556 (smallest primitive prime factor of Fibonacci(n), Lucas(n) and Pell(n)).

Cf. A086597, A086600 (number of primitive prime factors in Fibonacci(n) and Lucas(n)).

cp's OEIS Frontend

A172115 Partial sums of A001605.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Perl

A285314 Numbers k such that the k-th term of some (generalized) Lucas sequence has no primitive prime factor.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs