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
KEYWORD
nonn
AUTHOR
Michael Kaltman, May 31 2015
STATUS
approved