OFFSET
1,1
COMMENTS
From Andrew Howroyd, Dec 22 2024: (Start)
For any prime p, there are finitely many x such that x^2 - 2 has p as its largest prime factor.
The Filip Najman data file gives all 537 numbers x such that x^2 - 2 has no prime factor greater than 199. This includes a value for x = 1 which is not included here. (End)
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..536 (first 21 rows for primes up to 199)
Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355.
Filip Najman, List of Publications Page (Adjacent to entry number 7 are links with a data file for the first 21 rows of this sequence).
EXAMPLE
Triangle of numbers k such that p is the greatest prime factor of k^2 - 2:
p\k | 1 | 2 | 3 | 4 | 5 | 6 | 7 | >= 8
------------------------------------------------------------------------
2 | 2 | | | | | | |
7 | 3 | 4 | 10 | | | | |
17 | 6 | 11 | 45 | 108 | | | |
23 | 5 | 18 | 28 | 74 | 156 | 235 | |
31 | 8 | 23 | 39 | 116 | 1201 | | |
41 | 17 | 24 | 58 | 147 | 304 | 550 | 2272 | 390050;
47 | 7 | 40 | 54 | 87 | 101 | 181 | 557 | 1558, 43764, 314766;
71 | 12 | 59 | 130 | 225 | 414 | 1077 | 1124 | 2686, 3420, 4035;
73 | 32 | 41 | 178 | 333 | 698 | 844 | 1638 | 4567, 15362, 364384;
...
6 is a term of row 3 because (6^2 - 2)/17 = 2 and 2 < 17;
11 is a term of row 3 because (11^2 - 2)/17 = 7 and 7 < 17;
45 is a term of row 3 because (45^2 - 2)/17^2 = 7 and 7 < 17;
108 is a term of row 3 because (108^2 - 2)/17 = 686 = 2*7^3 and 7 < 17.
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Juri-Stepan Gerasimov, May 16 2014
EXTENSIONS
Converted to triangle by Andrew Howroyd, Dec 22 2024
STATUS
approved