A088232 - OEIS

%I #36 May 18 2022 10:07:08

%S 1,2,3,4,5,6,8,10,11,12,15,16,17,20,22,23,24,25,29,30,32,33,34,40,41,

%T 44,46,47,48,50,51,53,55,58,59,60,64,66,68,69,71,75,80,82,83,85,87,88,

%U 89,92,94,96,100,101,102,106,107,110,113,115,116,118,120,121,123,125,128

%N Numbers k such that 3 does not divide phi(k).

%C n such that the congruence x^3 == 1 mod(n) has only the trivial solution x=1 i.e. A060839(n) = 1 . Complement of sequence A066498.

%C Let U(n) be the group of positive integers coprime to n under mod n multiplication.  Let U(n)^3 = {x^3: x is an element of U(n)}.  These are the n such that U(n) = U(n)^3. - _Geoffrey Critzer_, Jun 07 2015

%C In other words, numbers divisible neither by 9 nor by any primes of the form 6k+1. - _Ivan Neretin_, Sep 03 2015

%C The asymptotic density of this sequence is 0 (Dressler, 1975). - _Amiram Eldar_, Jul 23 2020

%H Charles R Greathouse IV, <a href="/A088232/b088232.txt">Table of n, a(n) for n = 1..10000</a>

%H Robert E. Dressler, <a href="http://www.numdam.org/item/?id=CM_1975__31_2_115_0">A property of the phi and sigma_j functions</a>, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.

%H Kevin Ford, Florian Luca, and Pieter Moree, <a href="http://arxiv.org/abs/1108.3805">Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields</a>, arXiv:1108.3805 [math.NT], 2011-2012.

%F a(n) ~ k n sqrt(log(n)) for some constant k. k appears to be around 1.08. [_Charles R Greathouse IV_, Feb 14 2012]

%p select(t -> numtheory:-phi(t) mod 3 <> 0, [$1..1000]); # _Robert Israel_, Sep 04 2015

%t Prepend[Position[Table[Union[Select[Range[n], CoprimeQ[#, n] &]] ==

%t      Union[Mod[Select[Range[n], CoprimeQ[#, n] &]^3, n]], {n, 1,155}], True], 1] // Flatten (* _Geoffrey Critzer_, Jun 07 2015 *)

%t Select[Range[140],!Divisible[EulerPhi[#],3]&] (* _Harvey P. Dale_, Sep 23 2017 *)

%o (PARI) is(n)=eulerphi(n)%3 \\ _Charles R Greathouse IV_, Feb 04 2013

%Y Cf. A000010, A066498 (complement).

%Y Positions of 1's in A060839, of 0's in A354099, of nonzeros in A074942.

%Y Cf. also A329963.

%K nonn

%O 1,2

%A Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 03 2003

%E More terms from _Ray Chandler_, Nov 05 2003