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A256011 Integers n with the property that the largest prime factor of n^2+1 is less than n. 5
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 (list; graph; refs; listen; history; text; internal format)
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.

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

CROSSREFS

Cf. A002144 (Pythagorean primes), A014442.

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

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Last modified May 21 15:12 EDT 2019. Contains 323444 sequences. (Running on oeis4.)