A248527 Numbers n such that the smallest prime divisor of n^2+1 is 13.

34, 44, 60, 70, 86, 96, 164, 174, 190, 200, 216, 226, 294, 304, 320, 330, 346, 356, 424, 434, 450, 460, 476, 486, 554, 564, 580, 590, 606, 616, 684, 694, 710, 720, 736, 746, 814, 824, 840, 850, 866, 876, 944, 954, 970, 980, 996, 1006, 1074, 1084, 1100, 1110

Offset: 1

Author

Michel Lagneau, Oct 08 2014

Keywords

• nonn

Comments

Or numbers n such that the smallest prime divisor of A002522(n) is A002313(3).

a(n) == 8 (mod 26) if n is odd and a(n) == 18 (mod 26) if n is even.

It is interesting to observe that a(n) is given by a linear formula (see the formula below).

Examples

			34 is in the sequence because 34^2+1= 13*89.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002522, A089120, A002313.

Programs

Magma

[n: n in [2..3000] | PrimeDivisors(n^2+1)[1] eq 13]; // Bruno Berselli, Oct 08 2014

Maple

* first program *
with(numtheory):p:=13:
   for n from 1 to 1000 do:
    if factorset(n^2+1)[1] = p then printf(`%d, `, n):
    else
    fi:
   od:
* second program using the formula*
for n from 0 to 100 by 5 do:
   for k from 1 to 3 do:
     x:=8+(k+n)*26:y:=18+(k+n)*26:
     printf(`%d, `,x):printf(`%d, `,y):
   od:
  od:

Mathematica

lst={};Do[If[FactorInteger[n^2+1][[1,1]]==13,AppendTo[lst,n]],{n,2,2000}];lst
p = 13; ps = Select[Range[p - 1], Mod[#, 4] != 3 && PrimeQ[#] &]; Select[Range[1200], Divisible[(nn = #^2 + 1), p] && ! Or @@ Divisible[nn, ps] &] (* Amiram Eldar, Aug 16 2019 *)

PARI

isok(n) = factor(n^2+1)[1,1] == 13; \\ Michel Marcus, Oct 08 2014

Formula

{a(n)} = {8+(k + m)*26} union {18+(k + m)*26} for m = 0, 5, 10,...,5p,... and k = 1, 2, 3 (values in increasing order).