cp's OEIS Frontend

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

%I A167181 #45 Jul 21 2025 19:02:38
%S A167181 1,3,7,11,19,21,23,31,33,43,47,57,59,67,69,71,77,79,83,93,103,107,127,
%T A167181 129,131,133,139,141,151,161,163,167,177,179,191,199,201,209,211,213,
%U A167181 217,223,227,231,237,239,249,251,253,263,271,283,301,307,309,311,321,329
%N A167181 Squarefree numbers such that all prime factors are == 3 mod 4.
%C A167181 Or, numbers that are not divisible by the sum of two squares (other than 1). - Clarified by _Gabriel Conant_, Apr 18 2016
%C A167181 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
%H A167181 Robert Israel, <a href="/A167181/b167181.txt">Table of n, a(n) for n = 1..10000</a>
%H A167181 Rafael Jakimczuk, <a href="http://dx.doi.org/10.13140/RG.2.2.27745.48487">Generalizations of Mertens's Formula and k-Free and s-Full Numbers with Prime Divisors in Arithmetic Progression</a>, ResearchGate, 2024.
%H A167181 V Sai Prabhav, <a href="/A167181/a167181.pdf">Proof for the comment</a>
%F A167181 A005117 INTERSECT A004614. - _R. J. Mathar_, Nov 05 2009
%F A167181 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
%p A167181 N:= 1000: # to get all terms <= N
%p A167181 S:= {1};
%p A167181 for p from 3 by 4 to N do
%p A167181   if isprime(p) then
%p A167181     S:= S union select(`<=`, map(t -> t*p, S),N)
%p A167181   fi
%p A167181 od:
%p A167181 sort(convert(S,list)); # _Robert Israel_, Apr 18 2016
%t A167181 Select[Range@ 1000, #==1 || ({{3}, {1}} == Union /@ {Mod[ #[[1]], 4], #[[2]]} &@ Transpose@ FactorInteger@ #) &] (* _Giovanni Resta_, Apr 18 2016 *)
%o A167181 (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
%Y A167181 Cf. A002145, A004613, A004614, A005117, A231754.
%Y A167181 Cf. A175647, A243379.
%K A167181 nonn,easy
%O A167181 1,2
%A A167181 _Arnaud Vernier_, Oct 29 2009
%E A167181 Edited by _Zak Seidov_, Oct 30 2009
%E A167181 Narrowed definition down to squarefree numbers - _R. J. Mathar_, Nov 05 2009