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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 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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