Michael Kaltman - cp's OEIS frontend

User: Michael Kaltman

Michael Kaltman has authored 2 sequences.

A282538 Odd integers n with the property that the largest prime factor of n^2+4 is less than n.

11, 29, 49, 59, 99, 111, 121, 127, 141, 161, 179, 199, 205, 211, 213, 219, 237, 247, 261, 283, 289, 309, 311, 335, 359, 369, 387, 393, 411, 417, 419, 433, 441, 469, 479, 485, 521, 523, 527, 535, 569, 581, 595, 603, 611, 619, 621, 633, 643, 679, 691, 705, 711, 715, 723, 729, 739, 741, 749, 759

Offset: 1

Michael Kaltman, Feb 17 2017

nonn

Every Pythagorean prime p can be uniquely written as the sum of two positive integers a and b such that ab is congruent to 1 (mod p). If a>b, then the difference a-b must be an odd number; no number on this list can be said difference, and every positive odd integer NOT on this list is the difference of exactly one pair.

			Examples: 5 is not on this list, and 17-12=5 while 17+12=29 and (17)(12)==1 mod 29.  9 is not on this list, and 13-4=9 while 13+4=17 and (13)(4)==1 mod 17.  13 is not on this list, and 93-80=13 while 93+80=173 and (93)(80)==1 mod 173.  Note that 5^2+4=29, 9^2+4=85=17(5), and 13^2+4=173

Cf. A256011 (generated similarly, but for n^2+1 instead of n^2+4).

fQ[n_] := FactorInteger[n^2 + 4][[-1, 1]] < n; Select[2 Range[380] - 1, fQ] (* Robert G. Wilson v, Feb 17 2017 *)

isok(n) = (n%2) && vecmax(factor(n^2+4)[,1]) < n; \\ Michel Marcus, Feb 18 2017

a(22) onward from Robert G. Wilson v, Feb 17 2017

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

Original entry on oeis.org

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

Michael Kaltman, May 31 2015

nonn

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

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

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

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

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

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

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

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

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

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User: Michael Kaltman

A282538 Odd integers n with the property that the largest prime factor of n^2+4 is less than n.

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

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