A185389 Largest number k such that the greatest prime factor of k^2+1 is A002313(n), the n-th prime not congruent to 3 mod 4.

1, 7, 239, 268, 307, 18543, 2943, 485298, 330182, 478707, 24208144, 22709274, 2189376182, 284862638, 599832943, 19696179, 314198789, 3558066693, 69971515635443, 18986886768, 18710140581, 104279454193

Offset

1,2

Comments

For any prime p, there are finitely many k such that k^2+1 has p as its largest prime factor.

Numbers k such that k^2+1 is p-smooth appear in arctan-relations for the computation of Pi (for example, Machin's identity Pi/4 = 4*arctan(1/5) - arctan(1/239)), see the fxtbook link. [Joerg Arndt, Jul 02 2012]

Links

Table of n, a(n) for n=1..22.

Joerg Arndt, Matters Computational (The Fxtbook), section 32.5 "Arctangent relations for Pi", pp. 633-640.

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

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, Home Page (gives all 811 numbers x such that x^2+1 has no prime factor greater than 197)

Crossrefs

Equivalents for other polynomials: A175607 (k^2 - 1), A145606 (k^2 + k).

Sequence in context: A160491 A120661 A366703 * A159967 A349046 A139057

Adjacent sequences: A185386 A185387 A185388 * A185390 A185391 A185392

Keyword

nonn,hard,more

Author

Charles R Greathouse IV, Feb 21 2011

Status

approved