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 A076605 Largest prime divisor of n^2 - 1. 6
 3, 2, 5, 3, 7, 3, 7, 5, 11, 5, 13, 7, 13, 7, 17, 3, 19, 5, 19, 11, 23, 11, 23, 13, 5, 13, 29, 7, 31, 5, 31, 17, 11, 17, 37, 19, 37, 19, 41, 7, 43, 11, 43, 23, 47, 23, 47, 5, 17, 13, 53, 13, 53, 7, 19, 29, 59, 29, 61, 31, 61, 31, 13, 11, 67, 17, 67, 17, 71, 7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS Also the largest prime that divides either n-1 or n+1. Størmer shows that a(n) tends to infinity with n. Schinzel shows that lim inf a(n)/log log n >= 2 and, using lower bounds for linear forms of logarithms, this inequality can be generalized for general quadratic polynomials, with 2 replaced by 4/7 for irreducible ones and 2/7 for reducible ones. - Tomohiro Yamada, Apr 15 2017 REFERENCES K. Mahler, "Uber den grossten Primteiler spezieller Polynome zweiten Grades", Arch. Math. Naturvid. B.41, 1935, pp. 3 - 26. LINKS T. D. Noe, Table of n, a(n) for n = 2..10000 D. H. Lehmer, On a problem of Størmer, Ill. J. Math., 8 (1964), 57-79. A. Schinzel, On two theorems of Gelfond and some of their applications, Acta Arith. 13 (1967), 177--236. Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications (in French), Skrifter udgivne af Videnskabsselskabet i Christiania: Mathematisk-naturvidenskabelig Klasse, 2 (1897), 48 pp. EXAMPLE n=11: the largest prime factor of 10 and 12 is 5, therefore a(11) = 5. MATHEMATICA Table[ Last[ Table[ # [[1]]] & /@ FactorInteger[n^2 - 1]], {n, 2, 80}] PROG (PARI) for (n=3, 100, print1(", "max(factor(n-1)[, 1][length(factor(n-1)[, 1])], factor(n+1)[, 1][length(factor(n+1)[, 1])]))) CROSSREFS Cf. A006530, A037464, A074399 (bisections). Cf. A175607. Cf. A014442 (largest prime divisor of n^2 + 1). - Tomohiro Yamada, Apr 15 2017 Sequence in context: A124332 A246416 A165342 * A194748 A030640 A176447 Adjacent sequences:  A076602 A076603 A076604 * A076606 A076607 A076608 KEYWORD nonn AUTHOR Jon Perry, Oct 21 2002 STATUS approved

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Last modified May 26 03:36 EDT 2017. Contains 287073 sequences.