A216033 Numbers k such that every prime factor of k^2 + 1 is congruent to 1 (mod 8).

4, 16, 20, 24, 36, 40, 56, 64, 84, 100, 116, 120, 124, 140, 144, 156, 160, 176, 180, 184, 196, 204, 224, 236, 240, 256, 260, 264, 276, 280, 284, 296, 300, 324, 340, 344, 384, 396, 400, 404, 420, 436, 440, 444, 464, 480, 484, 496, 516, 536, 540, 544, 556, 576

Offset

1,1

Comments

From Robert Israel, Mar 29 2020: (Start)

All terms are divisible by 4.

Includes all terms of A005574 that are divisible by 4. (End)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

64 is in the sequence because 64^2 + 1 = 17*241 and {17, 241} == 1 (mod 8).

Maple

with(numtheory):for n from 1 to 1000 do:x:=factorset(n^2+1):n1:=nops(x):s1:=0:for m from 1 to n1 do: if irem(x[m], 8)=1 then s1:=s1+1:else fi:od:if s1=n1 then printf(`%d, `, n):else fi:od:
# Alternative:
select(n -> numtheory:-factorset(n^2+1) mod 8 = {1}, 4*[$1..1000]); # Robert Israel, Mar 29 2020

Mathematica

Select[Range[576], Union[Mod[Transpose[FactorInteger[#^2 + 1]][[1]], 8]] == {1} &] (* T. D. Noe, Aug 31 2012 *)
Select[Range[600], AllTrue[FactorInteger[#^2+1][[All, 1]], Mod[#, 8]==1&]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 31 2021 *)

Prog

(Magma) [n: n in [1..600] | forall{PrimeDivisors(n^2+1)[i]: i in [1..#PrimeDivisors(n^2+1)] | IsOne(PrimeDivisors(n^2+1)[i] mod 8)}]; // Bruno Berselli, Aug 30 2012

Crossrefs

Cf. A005574, A216032.

Sequence in context: A328465 A280844 A277887 * A071966 A349521 A326781

Adjacent sequences: A216030 A216031 A216032 * A216034 A216035 A216036

Keyword

nonn

Author

Michel Lagneau, Aug 30 2012

Status

approved