|
%I #49 Sep 15 2023 07:52:32
%S 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,
%T 29,29,31,31,11,17,17,7,37,37,19,13,41,41,43,43,11,23,47,47,7,7,17,17,
%U 53,53,11,11,19,29,59,59,61,61,31,7,13,13,67,67,23,23,71,71,73,73,37,19
%N a(n) is the largest prime divisor of n(n+1).
%C Størmer shows that a(n) tends to infinity with n. Pólya generalized this result to other polynomials.
%C Kotov shows that a(n) >> log log n. - _Charles R Greathouse IV_, Mar 26 2012
%C 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
%C 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
%D S. V. Kotov, The greatest prime factor of a polynomial (in Russian), Mat. Zametki 13 (1973), pp. 515-522.
%D K. Mahler, Über den größten Primteiler spezieller Polynome zweiten Grades, Archiv for mathematik og naturvidenskab 41:6 (1934), pp. 3-26.
%D Georg Pólya, Zur arithmetischen Untersuchung der Polynome, Math. Zeitschrift 1 (1918), pp. 143-148.
%H Charles R Greathouse IV, <a href="/A074399/b074399.txt">Table of n, a(n) for n = 1..10000</a>
%H P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1976-39.pdf">Problems and results on number theoretic properties of consecutive integers and related questions</a>, 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.
%H M. Keates, <a href="https://doi.org/10.1017/S0013091500012967">On the greatest prime factor of a polynomial</a> (1968), pp. 301-303.
%H A. Schinzel, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa13/aa13113.pdf">On two theorems of Gelfond and some of their applications</a>, Acta Arithmetica 13:2 (1967-1968), pp. 177-236.
%H Carl Størmer, <a href="http://www.archive.org/stream/skrifterudgivnea1897chri#page/n79/mode/2up">Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications</a> (in French), Skrifter udgivne af Videnskabsselskabet i Christiania: Mathematisk-naturvidenskabelig Klasse (1897).
%F a(n) = Max (A006530(2n), A006530(2n+2)).
%t Table[ Last[ Table[ # [[1]]] & /@ FactorInteger[n^2 - 1]], {n, 3, 160, 2}]
%t Table[FactorInteger[n(n+1)][[-1,1]],{n,80}] (* _Harvey P. Dale_, Sep 28 2021 *)
%o (PARI) gpf(n)=my(f=factor(n)[,1]); f[#f]
%o a(n)=if(n<3, n+1, max(gpf(n),gpf(n+1))) \\ _Charles R Greathouse IV_, Sep 14 2015
%Y With A037464, the bisections of A076605.
%Y Essentially the same as A069902.
%Y Positions of primes <= p: A085152 (p=5), A085153 (p=7), A252494 (p=11), A252493 (p=13), A252492 (p=17).
%Y Last position of each prime: A002072.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_, Nov 29 2002
%E Extended by _Robert G. Wilson v_, Dec 02 2002
|
