A037464 -id:A037464 - cp's OEIS frontend

A076605 Largest prime divisor of n^2 - 1.

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

Offset: 2

Jon Perry, Oct 21 2002

nonn

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

			n=11: the largest prime factor of 10 and 12 is 5, therefore a(11) = 5.

K. Mahler, "Uber den grossten Primteiler spezieller Polynome zweiten Grades", Arch. Math. Naturvid. B.41, 1935, pp. 3 - 26.

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.

Cf. A006530, A037464, A074399 (bisections).
Cf. A175607.
Cf. A014442 (largest prime divisor of n^2 + 1). - Tomohiro Yamada, Apr 15 2017

Table[ Last[ Table[ # [[1]]] & /@ FactorInteger[n^2 - 1]], {n, 2, 80}]

for (n=3,100, print1(","max(factor(n-1)[,1][length(factor(n-1)[,1])],factor(n+1)[,1][length(factor(n+1)[,1])])))

A074399 a(n) is the largest prime divisor of n(n+1).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 29 2002

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

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.

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).

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.

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 *)

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

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

Extended by Robert G. Wilson v, Dec 02 2002

A321069 Greatest prime factor of n^3+2.

Original entry on oeis.org

3, 5, 29, 11, 127, 109, 23, 257, 43, 167, 43, 173, 733, 1373, 307, 683, 983, 2917, 2287, 4001, 157, 71, 283, 223, 5209, 47, 127, 3659, 24391, 587, 9931, 113, 433, 6551, 809, 569, 307, 27437, 433, 10667, 439, 239, 1559, 223, 91127, 16223, 4153, 457, 39217, 62501

Offset: 1

Charles R Greathouse IV, Oct 27 2018

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
D. R. Heath-Brown, The largest prime factor of x^3+2, Proceedings of the London Mathematical Society, 82:3 (2000), pp. 554-596.
Christopher Hooley, On the greatest prime factor of a cubic polynomial, Journal für die reine und angewandte Mathematik, 303 (1978), pp. 21-50.
A. J. Irving, The largest prime factor of x^3+2, arXiv:1412.0024 [math.NT], 2014.

Greatest prime factors of polynomials: A006530 (n), A076565 (2n+1), A076566 (3n+3), A076567 (4n+6), A164314 (n^2-2), A076605 (n^2-1), A014442 (n^2+1), A069902 (n^2+n), A074399 (n^2+n), A199423 (2n^2+n), A089619 (2n^2+2n+1), A037464 (4n^2-1), A253254 (9n^2-7n), A093074 (n^3-n), A081257 (n^3-1), A081256 (n^3+1), A321069(n^3+2), A281793 (n^3+n^2+n+1), A281793 (n^4-1), A096172 (n^4+1), A190136 (n^4 + 6n^3 + 11n^2 + 6n), A140538 (2n^4+1), A240548 (n^5+1), A281794 (n^5+n^3+n^2+1), A240549 (n^6+1), A240550 (n^7+1), A240551 (n^8+1), A240552 (n^9+1), A240553 (n^10+1).

[Maximum(PrimeDivisors(n^3 + 2)): n in [1..60]]; // Vincenzo Librandi, Oct 27 2018

Table[FactorInteger[n^3 + 2] [[-1, 1]], {n, 80}] (* Vincenzo Librandi, Oct 27 2018 *)

a(n) = vecmax(factor(n^3+2)[,1]); \\ Michel Marcus, Oct 27 2018

cp's OEIS Frontend

A076605 Largest prime divisor of n^2 - 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

A074399 a(n) is the largest prime divisor of n(n+1).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A321069 Greatest prime factor of n^3+2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI