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A256011
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Integers n with the property that the largest prime factor of n^2 + 1 is less than n.
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9
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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
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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7^2 + 1 = 50 = 2 * 5^2;
18^2 + 1 = 325 = 5^2 * 13;
21^2 + 1 = 442 = 2 * 13 * 17.
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MAPLE
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select(n -> max(numtheory:-factorset(n^2+1))<n, [$1..10^4]); # Robert Israel, Jun 09 2015
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MATHEMATICA
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Select[Range[10^4], FactorInteger [#^2 + 1][[-1, 1]] < # &] (* Giovanni Resta, Jun 09 2015 *)
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PROG
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(PARI) for(n=1, 10^3, N=n^2+1; if(factor(N)[, 1][omega(N)] < n, print1(n, ", "))) \\ Derek Orr, Jun 08 2015
(Magma) [k:k in [1..330]| Max(PrimeDivisors(k^2+1)) lt k]; // Marius A. Burtea, Jul 27 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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