A242747 Largest number k such that (k!+n!)/(k+n) is an integer.

2, 8, 25, 2874, 12337, 125124, 40312, 362871, 119710813081, 226288998, 479001588, 379491816246, 2857509238543899, 284854629849752642, 20922789887984, 158339857954376933898893600, 7802547932734125256832521, 1785042244078013092809522, 2432902008176639980

Offset

2,1

Comments

For n > 2, the conjectured second-to-last value of k that makes (k!+n!)/(k+n) an integer is k = n!-n. It is believed that for n > 2, there is a formula for the last value, a(n).

Also, for n > 2, k+n is prime. Thus (k-n+1)!+1 (by Wilson's Theorem) and k!+n! are both multiples of k+n. Further, it is conjectured that the smallest prime factor of k!/n!+1 is k+n.

Based on the formula given by Hiroaki Yamanouchi, k+n need not be prime if n! > gpf(n!*(n-1)! + (-1)^n). - Derek Orr, Sep 26 2014

Links

Hiroaki Yamanouchi, Table of n, a(n) for n = 2..34

Formula

a(n) = max(gpf(n!*(n-1)! + (-1)^n), n!) - n (for n >= 3), where gpf(x) is the greatest prime factor of x. - Hiroaki Yamanouchi, Sep 26 2014

Example

(8!+3!)/(8+3) = 3666 is an integer. There are no other integers > 8 that make this quotient an integer. Thus a(3) = 8.

Prog

(PARI) a(n) = if(n==2, 2, max(vecmax(factor(n!*(n-1)!-2*(n%2)+1)[, 1]), n!) - n); \\ Modified by Jinyuan Wang, Mar 13 2020

Crossrefs

Cf. A006530.

Sequence in context: A150671 A150672 A150673 * A195585 A102942 A212691

Adjacent sequences: A242744 A242745 A242746 * A242748 A242749 A242750

Keyword

nonn

Author

Derek Orr, May 21 2014

Extensions

a(8)-a(20) from Hiroaki Yamanouchi, Sep 26 2014

Status

approved