A164314 -id:A164314 - cp's OEIS frontend

A242488 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.

2, 3, 4, 10, 6, 11, 45, 108, 5, 18, 28, 74, 156, 235, 8, 23, 39, 116, 1201, 17, 24, 58, 147, 304, 550, 2272, 390050, 7, 40, 54, 87, 101, 181, 557, 1558, 43764, 314766, 12, 59, 130, 225, 414, 1077, 1124, 2686, 3420, 4035, 32, 41, 178, 333, 698, 844, 1638, 4567, 15362, 364384

Offset: 1

Juri-Stepan Gerasimov, May 16 2014

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)

			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.

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).

Cf. A038873, A164314, A059770 (first terms for n>1), A185396 (last terms), A379348 (row lengths).

Cf. A223701.

Converted to triangle by Andrew Howroyd, Dec 22 2024

A363102 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(-2))))).

Original entry on oeis.org

7, 7, 23, 17, 47, 31, 79, 7, 17, 71, 167, 97, 223, 127, 41, 23, 359, 199, 439, 241, 31, 41, 89, 337, 727, 1, 839, 449, 137, 73, 1087, 577, 1223, 647, 1367, 103, 1, 47, 73, 881, 1847, 967, 1, 151, 2207, 1151, 2399, 1249, 113, 193, 401, 1, 3023, 1567, 191, 41, 71, 257, 3719, 113, 3967, 89, 103, 311

Offset: 3

Mohammed Bouras, May 19 2023

nonn

Conjecture 1: The sequence contains only 1's and primes.
Conjecture 2: All prime numbers appear either twice (same as A356247 and A357127) or three times.
Similar terms of A164314.
Conjecture: Record values correspond to A028871(m), m > 1. - Bill McEachen, Mar 06 2024
a(n) = 1 positions appear to correspond to A060515(m), m > 2. - Bill McEachen, Aug 05 2024

			a(5) = (5^2 - 2)/gcd(5^2 - 2, 2*A051403(5-3) + 5*A051403(5-4))= 23.
a(6) = a(11) = 6 + 11 = 17.
a(7) = a(40) = 7 + 40 = 47.

Bill McEachen, Table of n, a(n) for n = 3..10002
Mohammed Bouras, The Distribution Of Prime Numbers And Continued Fractions, (ppt) (2022)

Cf. A008865, A051403, A059772, A164314, A356247, A357127.

a051403(n) = (n+2)*sum(k=0, n, k!)/2;
a(n) = (n^2 - 2)/gcd(n^2 - 2, 2*a051403(n-3) + n*a051403(n-4)); \\ Michel Marcus, May 24 2023

a(n) = (n^2 - 2)/gcd(n^2 - 2, 2*A051403(n-3) + n*A051403(n-4)).
a(n) = A164314(n) if A164314(n) > n.
If a(n) = a(m) and n < m < a(n), then a(n) = n + m.

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

A242488 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.

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A363102 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(-2))))).

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A321069 Greatest prime factor of n^3+2.

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