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A336993 List of m such that b(m) has no primitive factor, where {b(m)} is the generalized Lucas sequence defined by b(0) = 0, b(1) = 1 and b(m) = b(m-1) - 2*b(m-2) for m >= 2 (A107920). 0
1, 2, 3, 5, 8, 12, 13, 18, 30 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A generalized Lucas sequence {U(m)} is a sequence defined by U(0) = 0, U(1) = 1 and U(m) = P*U(m-1) - Q*U(m-2) for m >= 2, where P > 0, gcd(P, Q) = 1 and (P, Q) != (1, 0), (1, 1) or (2, 1). The discriminant is D = P^2 - 4*Q.

A primitive factor of U(m), m > 0 is any prime that divides U(m) but does not divide U(r) for any 0 < r < m. Let's call a prime factor of U(m) "strongly primitive" if it is primitive and does not divide D.

Bilu, Hanrot and Voutier shows that for (P, Q, D) != (1, 2, -7), if U(m) has no strongly primitive factor, than m = 1..8, 10 or 12 (except for a few cases, m can only be 1, 2, 3, 4 or 6. See A285314). (P, Q, D) = (1, 2, -7) is the only case where U(13), U(18) and U(30) have no primitive factor.

This is also the only case where there are nine terms without a primitive factor. The case (P, Q, D) = (1, 3, -11) has five (m = 1, 2, 5, 6, 12). (1, -1, 5) has four (m = 1, 2, 6, 12). (1, 5, -19) has four (m = 1, 2, 7, 12). For all other Lucas sequences there can be at most three terms without a primitive factor (e.g. m = 1, 2, 6 for (3, 2, 1)).

Note that b(7) = 7 has only a primitive factor (7) that is not strongly primitive.

LINKS

Table of n, a(n) for n=1..9.

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.

EXAMPLE

We have b(1) = b(2) = 1 and b(3) = b(5) = b(13) = -1, so obviously b(m) has no primitive factor if m = 1, 2, 3, 5, 13.

b(8) = -3 has only one prime factor 3, but 3 divides b(4) = -3, so 8 is a term here.

b(12) = 45 has two prime factors 3 and 5, but 3 divides b(4) = -3 and 5 divides b(6) = 5, so 12 is here.

b(18) = 85 has two prime factors 5 and 17, but 5 divides b(6) = 5 and 17 divides b(9) = -17, so 18 is here.

b(30) = -24475 has three prime factors 5, 11 and 89, but 5 divides b(6) = 5, 11 divides b(10) = -11 and 89 divides b(15) = -89, so 30 is also here.

According to Bilu, Hanrot and Voutier, b(m) has at least one primitive factor for any other m (and at least one strongly primitive factor if m != 7).

CROSSREFS

Cf. A285314, A107920.

Sequence in context: A020899 A057987 A243165 * A176746 A067288 A308075

Adjacent sequences:  A336990 A336991 A336992 * A336994 A336995 A336996

KEYWORD

nonn,fini,full

AUTHOR

Jianing Song, Aug 10 2020

STATUS

approved

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Last modified September 29 18:27 EDT 2020. Contains 337432 sequences. (Running on oeis4.)