A235866 - OEIS

%I #23 Aug 21 2021 16:34:50

%S 1,3,5,7,11,13,15,17,19,21,23,29,31,35,37,41,43,47,51,53,55,57,59,61,

%T 65,67,71,73,77,79,83,85,87,89,91,93,97,101,103,105,107,109,113,115,

%U 119,123,127,129,131,133,137,139,143,145,149,151,155,157,159,161

%N G-cyclic numbers: numbers n such that gcd(n,A060968(n))=1.

%C From _Robert Israel_, May 01 2020: (Start)

%C All terms are odd and squarefree.

%C Contains all odd primes.

%C If n is a member, then so are all divisors of n.

%C (End)

%H Robert Israel, <a href="/A235866/b235866.txt">Table of n, a(n) for n = 1..10000</a>

%H Jose María Grau, A. M. Oller-Marcen, Manuel Rodriguez and D. Sadornil, <a href="http://arxiv.org/abs/1401.4708">Fermat test with Gaussian base and Gaussian pseudoprimes</a>, arXiv:1401.4708 [math.NT], 2014.

%p g:= proc(p,e) if p=2 or e > 1 then 0

%p   elif p mod 4 = 1 then p-1 else p+1 fi

%p end proc:

%p h:= proc(n) mul(g(t[1],t[2]),t=ifactors(n)[2]) end proc:

%p select(n -> igcd(n,h(n))=1, [seq(i,i=1..2000,2)]); # _Robert Israel_, May 01 2020

%t fa=FactorInteger; phi[1]=1;phi[p_, s_] := Which[Mod[p, 4] == 1, p^(s-1)*(p-1), Mod[p, 4]==3, p^(s-1)*(p+1), s==1, 2, True, 2^(s+1)]; phi[1]=1; phi[n_] := Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i, Length[fa[n]]}]; Select[Range[1000], GCD[phi[#], #] == 1 &]

%o (PARI) genit(maxx)={arr=List(); for(ptr=1, maxx, if(gcd(ptr,A060968(ptr))==1, listput(arr,ptr))); arr}

%o \\******** following code taken from A060968

%o A060968(n)={my(f=factor(n)[,1]); q=n*prod(i=if(n%2,1,2),#f,if(f[i]%4==1,1-1/f[i],1+1/f[i]))*if(n%4,1,2);return(q);} \\ _Bill McEachen_, Jul 16 2021

%Y Cf. A060968, A003277.

%K nonn

%O 1,2

%A _José María Grau Ribas_, Jan 19 2014