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%I #24 Dec 22 2024 23:56:33
%S 2,3,4,10,6,11,45,108,5,18,28,74,156,235,8,23,39,116,1201,17,24,58,
%T 147,304,550,2272,390050,7,40,54,87,101,181,557,1558,43764,314766,12,
%U 59,130,225,414,1077,1124,2686,3420,4035,32,41,178,333,698,844,1638,4567,15362,364384
%N Triangle read by rows in which row n lists numbers k such that the greatest prime factor of k^2 - 2 is A038873(n), the n-th prime not congruent to 3 or 5 mod 8.
%C From _Andrew Howroyd_, Dec 22 2024: (Start)
%C For any prime p, there are finitely many x such that x^2 - 2 has p as its largest prime factor.
%C 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)
%H Andrew Howroyd, <a href="/A242488/b242488.txt">Table of n, a(n) for n = 1..536</a> (first 21 rows for primes up to 199)
%H Filip Najman, <a href="http://web.math.hr/~fnajman/smooth.pdf">Smooth values of some quadratic polynomials</a>, Glasnik Matematicki Series III 45 (2010), pp. 347-355.
%H Filip Najman, <a href="https://web.math.pmf.unizg.hr/~fnajman/publications.html">List of Publications Page</a> (Adjacent to entry number 7 are links with a data file for the first 21 rows of this sequence).
%e Triangle of numbers k such that p is the greatest prime factor of k^2 - 2:
%e p\k | 1 | 2 | 3 | 4 | 5 | 6 | 7 | >= 8
%e ------------------------------------------------------------------------
%e 2 | 2 | | | | | | |
%e 7 | 3 | 4 | 10 | | | | |
%e 17 | 6 | 11 | 45 | 108 | | | |
%e 23 | 5 | 18 | 28 | 74 | 156 | 235 | |
%e 31 | 8 | 23 | 39 | 116 | 1201 | | |
%e 41 | 17 | 24 | 58 | 147 | 304 | 550 | 2272 | 390050;
%e 47 | 7 | 40 | 54 | 87 | 101 | 181 | 557 | 1558, 43764, 314766;
%e 71 | 12 | 59 | 130 | 225 | 414 | 1077 | 1124 | 2686, 3420, 4035;
%e 73 | 32 | 41 | 178 | 333 | 698 | 844 | 1638 | 4567, 15362, 364384;
%e ...
%e 6 is a term of row 3 because (6^2 - 2)/17 = 2 and 2 < 17;
%e 11 is a term of row 3 because (11^2 - 2)/17 = 7 and 7 < 17;
%e 45 is a term of row 3 because (45^2 - 2)/17^2 = 7 and 7 < 17;
%e 108 is a term of row 3 because (108^2 - 2)/17 = 686 = 2*7^3 and 7 < 17.
%Y Cf. A038873, A164314, A059770 (first terms for n>1), A185396 (last terms), A379348 (row lengths).
%Y Cf. A223701.
%K nonn,tabf
%O 1,1
%A _Juri-Stepan Gerasimov_, May 16 2014
%E Converted to triangle by _Andrew Howroyd_, Dec 22 2024