A223702 Irregular triangle of numbers k such that A002313(n), the n-th prime not congruent to 3 mod 4 is the largest prime factor of k^2 + 1.

1, 2, 3, 7, 5, 8, 18, 57, 239, 4, 13, 21, 38, 47, 268, 12, 17, 41, 70, 99, 157, 307, 6, 31, 43, 68, 117, 191, 302, 327, 882, 18543, 9, 32, 73, 132, 278, 378, 829, 993, 2943, 23, 30, 83, 182, 242, 401, 447, 606, 931, 1143, 1772, 6118, 34208, 44179, 85353, 485298

Offset

1,2

Comments

Note that primes of the form 4x+3 are not divisors.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..811 (first 22 rows for primes up to 197)

Florian Luca, Primitive divisors of Lucas sequences and prime factors of x^2 + 1 and x^4 + 1, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004), pp. 19-24.

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 22 rows (=811 terms) of this sequence). [As of Dec 2024, the link has the incorrect URL. Should be https://web.math.pmf.unizg.hr/~fnajman/rezplus1.html]

Example

Irregular triangle:
   p | {k}
-----+---------------------------------
   2 | {1},
   5 | {2, 3, 7},
  13 | {5, 8, 18, 57, 239},
  17 | {4, 13, 21, 38, 47, 268},
  29 | {12, 17, 41, 70, 99, 157, 307},
  37 | {6, 31, 43, 68, 117, 191, 302, 327, 882, 18543},
  41 | {9, 32, 73, 132, 278, 378, 829, 993, 2943}
  ...

Mathematica

t = Table[FactorInteger[n^2 + 1][[-1, 1]], {n, 10^5}]; Table[Flatten[Position[t, Prime[n]]], {n, 13}]

Crossrefs

Cf. A002313, A014442, A177979 (first terms), A185389 (last terms), A223705, A285283, A379346 (row lengths), A379347 (row sums).

Cf. A285282, A285523.

Cf. A223701, A223703, A223704 (related tables).

Sequence in context: A225403 A069786 A128804 * A120726 A060203 A131880

Adjacent sequences: A223699 A223700 A223701 * A223703 A223704 A223705

Keyword

nonn,tabf

Author

T. D. Noe, Apr 03 2013

Extensions

Definition amended by Andrew Howroyd, Dec 22 2024

Status

approved