A074399 - OEIS

A074399

a(n) is the largest prime divisor of n(n+1).

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.

Hector Pasten, The largest prime factor of n^2+1 and improvements on subexponential ABC, arXiv:2312.03566 [math.NT] (2024)

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)).

Pasten proves that a(n) >> (log log n)^2/(log log log n), see Corollary 1.5. - Charles R Greathouse IV, Oct 14 2024

MATHEMATICA

Table[ Last[ Table[ # [[1]]] & /@ FactorInteger[n^2 - 1]], {n, 3, 160, 2}]

Table[FactorInteger[n(n+1)][[-1, 1]], {n, 80}] (* Harvey P. Dale, Sep 28 2021 *)

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

With A037464, the bisections of A076605.

Essentially the same as A069902.

Positions of primes <= p: A085152 (p=5), A085153 (p=7), A252494 (p=11), A252493 (p=13), A252492 (p=17).

Last position of each prime: A002072.

Sequence in context: A185075 A342693 A349338 * 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