A379346 -id:A379346 - cp's OEIS frontend

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

T. D. Noe, Apr 03 2013

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

			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}
  ...

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]

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).

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

Definition amended by Andrew Howroyd, Dec 22 2024

A285283 Number of integers x such that the greatest prime factor of x^2 + 1 is at most A002313(n), the n-th prime not congruent to 3 mod 4.

Original entry on oeis.org

1, 4, 9, 15, 22, 32, 41, 57, 74, 94, 120, 156, 192, 232, 278, 325, 381, 448, 521, 607, 704, 811

Offset: 1

Tomohiro Yamada, Apr 16 2017

In other words, x^2 + 1 is A002313(n)-smooth.
Størmer shows that the number of such integers is finite for any n.
a(n) <= 3^n - 2^n follows from Størmer's argument.
a(n) <= (2^n-1)*(A002313(n)+1)/2 is implicit in Lehmer 1964.
Luca 2004 determines all integers x such that x^2 + 1 is 100-smooth, which is pushed to 200 by Najman 2010.

D. H. Lehmer, On a problem of Størmer, Ill. J. Math., 8 (1964), 57--69.
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, Glas. Mat. 45 (2010), 347--355. Tables are available in the author's Home Page (gives all 811 numbers x such that x^2+1 has no prime factor greater than 197).
A. Schinzel, On two theorems of Gelfond and some of their applications, Acta Arithmetica 13 (1967-1968), 177--236.
Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications (in French), Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. I (2), 48 pp.

Equivalents for x(x+1): A145604.

Cf. A002313, A014442, A185389, A223702, A285282, A379346 (first differences).

a(13)-a(22) added by Andrew Howroyd, Dec 22 2024

A379349 Number of integers of the form k^2 + 2 whose greatest prime factor is A033203(n), the n-th prime not congruent to 5 or 7 mod 8.

Original entry on oeis.org

1, 5, 5, 7, 11, 12, 18, 18, 21, 25, 30, 47, 39, 45, 62, 63, 83, 81, 107, 105, 130

Offset: 1

Andrew Howroyd, Dec 22 2024

See A379350 for additional information.

			Table showing n, p = A033203(n) and a(n):
   1    2    1
   2    3    5
   3   11    5
   4   17    7
   5   19   11
   6   41   12
   7   43   18
   8   59   18
   9   67   21
  10   73   25
  ...

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

Row lengths of A379350.

Cf. A033203, A181471, A379346, A379348.

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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.

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A285283 Number of integers x such that the greatest prime factor of x^2 + 1 is at most A002313(n), the n-th prime not congruent to 3 mod 4.

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A379349 Number of integers of the form k^2 + 2 whose greatest prime factor is A033203(n), the n-th prime not congruent to 5 or 7 mod 8.

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