A256011 Integers n with the property that the largest prime factor of n^2 + 1 is less than n.

7, 18, 21, 38, 41, 43, 47, 57, 68, 70, 72, 73, 83, 99, 111, 117, 119, 123, 128, 132, 133, 142, 157, 172, 173, 174, 182, 185, 191, 192, 193, 200, 211, 212, 216, 233, 237, 239, 242, 251, 253, 255, 265, 268, 273, 278, 293, 294, 302, 305, 307, 313, 319, 322, 327

Offset: 1

Author

Michael Kaltman, May 31 2015

Keywords

• nonn

Comments

Every Pythagorean prime, p, can be written as the sum of two positive integers, a and b, such that ab is congruent to 1 (mod p). Further: no number is the addend of two different primes, and the numbers that are NEVER addends are precisely the numbers in this list.

In particular: 5 = 2+3 and 2*3 = 6 == 1 (mod 5), 13 = 5+8 and 5*8 = 40 == 1 (mod 13), 17 = 4+13 and 4*13 = 52 == 1 (mod 17), 29 = 12+17 and 12*17 = 204 == 1 (mod 29), and so on.

Every integer greater than 1 is in exactly one of A002314, A152676, and the present sequence. - Michael Kaltman, May 11 2019

Examples

			7^2 + 1 = 50 = 2 * 5^2;
18^2 + 1 = 325 = 5^2 * 13;
21^2 + 1 = 442 = 2 * 13 * 17.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002144 (Pythagorean primes), A014442, A002314, A152676.

Programs

Magma

[k:k in [1..330]| Max(PrimeDivisors(k^2+1)) lt k]; // Marius A. Burtea, Jul 27 2019

Maple

select(n -> max(numtheory:-factorset(n^2+1))Robert Israel, Jun 09 2015

Mathematica

Select[Range[10^4], FactorInteger [#^2 + 1][[-1, 1]] < # &] (* Giovanni Resta, Jun 09 2015 *)

PARI

for(n=1,10^3,N=n^2+1;if(factor(N)[,1][omega(N)] < n,print1(n,", "))) \\ Derek Orr, Jun 08 2015

PARI

is(n)=my(f=factor(n^2+1)[,1]); f[#f]Charles R Greathouse IV, Jun 09 2015