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

Author

Tomohiro Yamada, Apr 22 2017

Keywords

• nonn
• fini
• full

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.

Examples

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

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.

Crossrefs

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

Programs

PARI

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