A167181 Squarefree numbers such that all prime factors are == 3 mod 4.

1, 3, 7, 11, 19, 21, 23, 31, 33, 43, 47, 57, 59, 67, 69, 71, 77, 79, 83, 93, 103, 107, 127, 129, 131, 133, 139, 141, 151, 161, 163, 167, 177, 179, 191, 199, 201, 209, 211, 213, 217, 223, 227, 231, 237, 239, 249, 251, 253, 263, 271, 283, 301, 307, 309, 311, 321, 329

Offset: 1

Author

Arnaud Vernier, Oct 29 2009

Keywords

• nonn
• easy

Comments

Or, numbers that are not divisible by the sum of two squares (other than 1). - Clarified by Gabriel Conant, Apr 18 2016

If a term divides the sum of two squares, then it divides each of the two numbers individually. Moreover, only the numbers in this sequence have this property. See link for proof. - V Sai Prabhav, Jul 15 2025

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Rafael Jakimczuk, Generalizations of Mertens's Formula and k-Free and s-Full Numbers with Prime Divisors in Arithmetic Progression, ResearchGate, 2024.
• V Sai Prabhav, Proof for the comment

Crossrefs

Cf. A002145, A004613, A004614, A005117, A231754.

Cf. A175647, A243379.

Programs

Maple

N:= 1000: # to get all terms <= N
S:= {1};
for p from 3 by 4 to N do
  if isprime(p) then
    S:= S union select(`<=`, map(t -> t*p, S),N)
  fi
od:
sort(convert(S,list)); # Robert Israel, Apr 18 2016

Mathematica

Select[Range@ 1000, #==1 || ({{3}, {1}} == Union /@ {Mod[ #[[1]], 4], #[[2]]} &@ Transpose@ FactorInteger@ #) &] (* Giovanni Resta, Apr 18 2016 *)

PARI

isok(n) = if (! issquarefree(n), return (0)); f = factor(n); for (i=1, #f~, if (f[i, 1] % 4 != 3, return (0))); 1 \\ Michel Marcus, Sep 04 2013

Formula

A005117 INTERSECT A004614. - R. J. Mathar, Nov 05 2009

The number of terms that do not exceed x is ~ c * x / sqrt(log(x)), where c = A243379/(2*sqrt(A175647)) = 0.4165140462... (Jakimczuk, 2024, Theorem 3.10, p. 26). - Amiram Eldar, Mar 08 2024

Extensions

Edited by Zak Seidov, Oct 30 2009

Narrowed definition down to squarefree numbers - R. J. Mathar, Nov 05 2009