A285523 Numbers n such that n^2 + 1 is 100-smooth

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 17, 18, 21, 22, 23, 27, 30, 31, 32, 34, 38, 41, 43, 46, 47, 50, 55, 57, 68, 70, 72, 73, 75, 83, 99, 117, 119, 123, 132, 133, 157, 172, 173, 182, 191, 216, 233, 239, 242, 255, 265, 268, 278, 302, 307, 319, 327, 378, 401, 411, 438, 447

Offset

1,2

Comments

Equivalently: Numbers n such that all prime factors of n^2 + 1 are <= 97.

Since an odd prime factor of n^2 + 1 must be of the form 4m + 1, n^2 + 1 must be of the form 2^t*5^a*13^b*17^c*29^d*37^e*41^f*53^g*61^h*73^i*89^j*97^k, with t = 0 or 1.

Luca determined all terms.

Links

Tomohiro Yamada, Table of n, a(n) for n = 1..156

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 n such that n^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 Nr. 2 (1897), 48 pp.

Example

157^2 + 1 = 2*5^2*17*29 so 157 is a term.

Prog

(PARI) isok(n) = vecmax(factor(n^2+1)[, 1]) <= 100; \\ Michel Marcus, Apr 23 2017

Crossrefs

Cf. A285282 (n^2 + 1 is 13-smooth), A285283.

Sequence in context: A246422 A072495 A257671 * A126968 A254785 A126969

Adjacent sequences: A285520 A285521 A285522 * A285524 A285525 A285526

Keyword

nonn,fini,full

Author

Tomohiro Yamada, Apr 22 2017

Status

approved