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 A074399 a(n) is the largest prime divisor of n(n+1). 10
 2, 3, 3, 5, 5, 7, 7, 3, 5, 11, 11, 13, 13, 7, 5, 17, 17, 19, 19, 7, 11, 23, 23, 5, 13, 13, 7, 29, 29, 31, 31, 11, 17, 17, 7, 37, 37, 19, 13, 41, 41, 43, 43, 11, 23, 47, 47, 7, 7, 17, 17, 53, 53, 11, 11, 19, 29, 59, 59, 61, 61, 31, 7, 13, 13, 67, 67, 23, 23, 71, 71, 73, 73, 37, 19 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Størmer shows that a(n) tends to infinity with n. Pólya generalized this result to other polynomials. Kotov shows that a(n) >> log log n. - Charles R Greathouse IV, Mar 26, 2012 Keates and Schinzel give effective constants for the above; in particular the latter shows that lim inf a(n)/log log n >= 2/7. - Charles R Greathouse IV, Nov 12 2012 Erdős conjectures ("on very flimsy probabilistic grounds") that for every e > 0, a(n) < (log n)^(2+e) infinitely often, while a(n) < (log n)^(2-e) only finitely often. - Charles R Greathouse IV, Mar 11 2015 REFERENCES S. V. Kotov, The greatest prime factor of a polynomial (in Russian), Mat. Zametki 13 (1973), pp. 515-522. K. Mahler, Über den größten Primteiler spezieller Polynome zweiten Grades, Archiv for mathematik og naturvidenskab 41:6 (1934), pp. 3-26. Georg Pólya, Zur arithmetischen Untersuchung der Polynome, Math. Zeitschrift 1 (1918), pp. 143-148. LINKS Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 P. Erdős, Problems and results on number theoretic properties of consecutive integers and related questions, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975), Congress. Numer. XVI , pp. 25-44, Utilitas Math., Winnipeg, Man., 1976. M. Keates, On the greatest prime factor of a polynomial (1968), pp. 301-303. A. Schinzel, On two theorems of Gelfond and some of their applications, Acta Arithmetica 13:2 (1967-1968), pp. 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 (1897). FORMULA a(n) = Max (A006530(2n), A006530(2n+2)). MATHEMATICA Table[ Last[ Table[ # []] & /@ FactorInteger[n^2 - 1]], {n, 3, 160, 2}] PROG (PARI) gpf(n)=my(f=factor(n)[, 1]); f[#f] a(n)=if(n<3, n+1, max(gpf(n), gpf(n+1))) \\ Charles R Greathouse IV, Sep 14 2015 CROSSREFS Bisection of A076605. Cf. A037464. Essentially the same as A069902. Cf. A085152, A085153, A252492, A252493, A252594 (all n's such that a(n) <= p for each given prime 5 <= p <= 17). Sequence in context: A247176 A325163 A185075 * A090302 A093074 A284412 Adjacent sequences:  A074396 A074397 A074398 * A074400 A074401 A074402 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Nov 29 2002 EXTENSIONS Extended by Robert G. Wilson v, Dec 02 2002 STATUS approved

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Last modified February 27 15:59 EST 2020. Contains 332307 sequences. (Running on oeis4.)