Abstract
In this paper, we study various arithmetic properties of the sequence (an)n≥1 satisfying the recurrence relation an = nan–1 + 1, n = 2, 3,..., with the initial term a1 = 0. In particular, we estimate the number of solutions of various congruences with this sequence and the number of distinct prime divisors of its first N terms.
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W. D. Banks and I. E. Shparlinski, On values taken by the largest prime factor of shifted primes, J. Austral. Math. Soc., 82 (2007), 133–147.
P. Cutter, A. Granville and T. J. Tucker, The number of fields generated by the square root of values of a given polynomial, Canad. Math. Bull., 46 (2003), 71–79.
M. Z. Garaev, F. Luca and I. E. Shparlinski, Character sums and congruences with n!, Trans. Amer. Math. Soc., 356 (2004), 5089–5102.
G. Everest, A. J. van der Poorten, I. E. Shparlinski and T. Ward, Recurrence sequences, Amer. Math. Soc., 2003.
H. Halberstam and H.-E. Richert, Sieve Methods, Academic Press, London, 1974.
F. Luca, Prime factors of Motzkin numbers, Ars Combin., 80 (2006), 87–96.
F. Luca, Prime divisors of binary holonomic sequences, Adv. in Appl. Math., to appear.
F. Luca and I. E. Shparlinski, Prime divisors of shifted factorials, Bull. London Math. Soc., 37 (2005), 809–817.
E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers II, Izv. Ross. Akad. Nauk. Ser. Mat., 64 (2000), 125–180; English translation: Izv. Math., 64 (2000), 1217–1269.
T. Müller, Prime and composite terms in Sloane’s sequence A056542, J. Integer Seq., 8 (2005), Article 05.3.3.
I. E. Shparlinski and D. Sutantyo, On the set of the largest prime divisors, Publ. Math. Debrecen, to appear.
I. E. Shparlinski and A. Winterhof, On the discrepancy and linear complexity of some counter-dependent recurrence sequences, Lecture Notes in Comp. Sci. 4086, Springer-Verlag, Berlin, 2006, 295–303.
N. J. A. Sloane, On-line encyclopedia of integer sequences, http://www.research.att.com/’njas/sequences.
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Communicated by Attila Pethő
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Balasuriya, S., Luca, F. & Shparlinski, I.E. Prime Divisors Of Some Recurrence Sequence. Period Math Hung 54, 215–227 (2007). https://doi.org/10.1007/s-10998-007-2215-z
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DOI: https://doi.org/10.1007/s-10998-007-2215-z