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.

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

Offset: 1

Author

Tomohiro Yamada, Apr 16 2017

Keywords

• nonn
• hard
• more

Comments

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.

Links

• 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.

Crossrefs

Equivalents for x(x+1): A145604.

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

Extensions

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