

A256011


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


7



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
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OFFSET

1,1


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


LINKS

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


EXAMPLE

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


MAPLE

select(n > max(numtheory:factorset(n^2+1))<n, [$1..10^4]); # Robert Israel, Jun 09 2015


MATHEMATICA

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


PROG

(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]<n \\ Charles R Greathouse IV, Jun 09 2015
(MAGMA) [k:k in [1..330] Max(PrimeDivisors(k^2+1)) lt k]; // Marius A. Burtea, Jul 27 2019


CROSSREFS

Cf. A002144 (Pythagorean primes), A014442, A002314, A152676.
Sequence in context: A185455 A103570 A223247 * A272972 A070609 A007236
Adjacent sequences: A256008 A256009 A256010 * A256012 A256013 A256014


KEYWORD

nonn


AUTHOR

Michael Kaltman, May 31 2015


STATUS

approved



