

A285283


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


3



1, 4, 9, 15, 22, 32, 41, 57, 74, 94, 120, 156
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OFFSET

1,2


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^n1)*(A002313(n)+1)/2 is implicit in Lehmer 1964.
Luca 2004 determines all integers x such that x^2 + 1 is 100smooth, which is pushed to 200 by Najman 2010.


LINKS

Table of n, a(n) for n=1..12.
D. H. Lehmer, On a problem of Størmer, Ill. J. Math., 8 (1964), 5769.
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. 1924.
Filip Najman, Smooth values of some quadratic polynomials, Glas. Mat. 45 (2010), 347355. 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 (19671968), 177236.
Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2  Dy^2 = +1 et leurs applications (in French), Skrifter Videnskabsselskabet (Christiania), Mat.Naturv. Kl. I (2), 48 pp.


CROSSREFS

Equivalents for x(x+1): A145604
Cf. A014442, A185389
Cf. A285282.
Sequence in context: A022443 A281026 A079423 * A243536 A184005 A194106
Adjacent sequences: A285280 A285281 A285282 * A285284 A285285 A285286


KEYWORD

nonn,hard,more


AUTHOR

Tomohiro Yamada, Apr 16 2017


STATUS

approved



