A185396 -id:A185396 - 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

A379348 Number of positive integers of the form k^2 - 2 whose greatest prime factor is A038873(n), the n-th prime not congruent to 3 or 5 mod 8.

Original entry on oeis.org

1, 3, 4, 6, 5, 8, 10, 10, 10, 14, 20, 22, 30, 30, 37, 42, 43, 48, 49, 64, 80

Offset: 1

Andrew Howroyd, Dec 22 2024

See A242488 for additional information.

			Table showing n, p = A038873(n) and a(n):
   1    2    1
   2    7    3
   3   17    4
   4   23    6
   5   31    5
   6   41    8
   7   47   10
   8   71   10
   9   73   10
  10   79   14
  ...

Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355.

Row lengths of A242488.

Cf. A038873, A185396, A379349.

A185397 Largest number x such that the greatest prime factor of x^2+2 is A033203(n), the n-th prime not congruent to 5 or 7 mod 8.

Original entry on oeis.org

22, 140, 707, 21362, 4991, 1306066, 137965, 2294636, 31768298, 1557652, 340064590, 38439662, 105080665, 273502688, 543164542, 9575480365630, 391890109484, 14629598023, 80849485336, 1241646894380

Offset: 1

Charles R Greathouse IV, Feb 21 2011

nonn

For any prime p, there are finitely many x such that x^2+2 has p as its largest prime factor.

Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355.
Filip Najman, Home Page (gives all 914 numbers x such that x^2+2 has no prime factor greater than 193)

Equivalents for other polynomials: A175607 (x^2 - 1), A145606 (x^2 + x), A185389 (x^2 + 1), A185396 (x^2 - 2).

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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A379348 Number of positive integers of the form k^2 - 2 whose greatest prime factor is A038873(n), the n-th prime not congruent to 3 or 5 mod 8.

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A185397 Largest number x such that the greatest prime factor of x^2+2 is A033203(n), the n-th prime not congruent to 5 or 7 mod 8.

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