A185397 -id:A185397 - cp's OEIS frontend

A379350 Triangle read by rows in which row n lists numbers k such that the greatest prime factor of k^2 + 2 is A033203(n), the n-th prime not congruent to 5 or 7 mod 8.

0, 1, 2, 4, 5, 22, 3, 8, 14, 19, 140, 7, 10, 24, 41, 58, 265, 707, 6, 13, 25, 32, 44, 63, 146, 184, 602, 3407, 21362, 11, 30, 52, 71, 112, 194, 298, 481, 503, 2695, 3433, 4991, 16, 27, 59, 70, 102, 113, 317, 500, 586, 1048, 2951, 3424, 4972, 8240, 12658, 83834, 686210, 1306066

Offset: 1

Andrew Howroyd, Dec 22 2024

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

			Irregular triangle begins:
   p | {k}
-----+------------------
   2 | {0}
   3 | {1, 2, 4, 5, 22}
  11 | {3, 8, 14, 19, 140}
  17 | {7, 10, 24, 41, 58, 265, 707}
  19 | {6, 13, 25, 32, 44, 63, 146, 184, 602, 3407, 21362}
  41 | {11, 30, 52, 71, 112, 194, 298, 481, 503, 2695, 3433, 4991}
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..915 (first 21 rows for primes up to 193)
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 rows 2..21 (=914 terms) of this sequence).

Cf. A033203, A379351, A379352 (first terms), A185397 (last terms), A379349 (row lengths).

Cf. A223701, A223702, A242488.

A379352 a(n) is the smallest nonnegative integer k such the greatest prime factor of k^2 + 2 is A033203(n), the n-th prime not congruent to 5 or 7 mod 8.

Original entry on oeis.org

0, 1, 3, 7, 6, 11, 16, 23, 20, 12, 9, 40, 17, 31, 26, 28, 51, 50, 18, 78, 34, 93, 15, 109, 38, 91, 68, 29, 127, 108, 130, 75, 141, 107, 46, 120, 143, 35, 96, 69, 21, 214, 37, 126, 94, 67, 163, 56, 190, 261, 216, 153, 239, 207, 260, 104, 43, 288, 62, 206, 77, 262, 64, 151, 346

Offset: 1

Andrew Howroyd, Dec 22 2024

nonn

			Table showing n, A033203(n), a(n), a(n)^2 + 2:
   1  2  0   2
   2  3  1   3
   3 11  3  11
   4 17  7  51 = 17*3
   5 19  6  38 = 19*2
   6 41 11 123 = 41*3
   7 43 16 258 = 43*3*2
   8 59 23 531 = 59*3^2
   9 67 20 402 = 67*3*2
  10 73 12 146 = 73*2
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A033203, A059100, A006530, A185397, A379350, A379351.

lista(n) = { my(L=List(),p=0); while(#L5&&r<>7, my(k=0); while(vecmax(factor(k^2 + 2)[,1]) <> p, k++); listput(L,k) )); Vec(L) }

cp's OEIS Frontend

A379350 Triangle read by rows in which row n lists numbers k such that the greatest prime factor of k^2 + 2 is A033203(n), the n-th prime not congruent to 5 or 7 mod 8.

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