cp's OEIS Frontend

A085397 Numbers that are not perfect powers and whose squarefree part is not congruent to 1 (mod 4).

%I A085397 #26 Feb 16 2025 08:32:50
%S A085397 2,3,6,7,10,11,12,14,15,18,19,22,23,24,26,28,30,31,34,35,38,39,40,42,
%T A085397 43,44,46,47,48,50,51,54,55,56,58,59,60,62,63,66,67,70,71,72,74,75,76,
%U A085397 78,79,82,83,86,87,88,90,91,92,94,95,96,98,99,102,103,104,106,107,108
%N A085397 Numbers that are not perfect powers and whose squarefree part is not congruent to 1 (mod 4).
%C A085397 Contains A016825. - _Robert Israel_, Mar 20 2016
%C A085397 The asymptotic density of this sequence is 2/3. - _Amiram Eldar_, Mar 09 2021
%H A085397 T. D. Noe, <a href="/A085397/b085397.txt">Table of n, a(n) for n=1..1000</a>
%H A085397 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ArtinsConstant.html">Artin's Constant</a>.
%p A085397 f:= proc(n) local F,x;
%p A085397   F:= ifactors(n)[2];
%p A085397   if igcd(seq(f[2],f=F)) > 1 then return false fi;
%p A085397   x:= mul(f[1], f = select(t -> t[2]::odd, F));
%p A085397   x mod 4 <> 1;
%p A085397 end proc:
%p A085397 select(f, [$1..200]); # _Robert Israel_, Mar 20 2016
%t A085397 fi[n_] := fi[n] = FactorInteger[n]; perfectPowerQ[n_] := Length[uf = Union[ fi[n][[All, 2]]]] == 1 && uf[[1]] >= 2; SquareFreePart[n_] := Times @@ Apply[Power, ({#[[1]], Mod[#[[2]], 2]} & ) /@ fi[n], {1}]; ok[n_] := ! perfectPowerQ[n] && Mod[ SquareFreePart[n], 4] != 1; Select[ Range[110], ok] (* _Jean-François Alcover_, Jan 20 2012 *)
%o A085397 (PARI) isok(n) = !ispower(n) && ((core(n) % 4) != 1); \\ _Michel Marcus_, Mar 19 2016
%Y A085397 Subsequence of A007916.
%Y A085397 Cf. A016825.
%K A085397 nonn,easy
%O A085397 1,1
%A A085397 _Eric W. Weisstein_, Jun 27 2003