%I #61 Sep 08 2022 08:44:58

%S 1,2,2,3,2,3,2,3,3,2,3,3,3,2,3,3,3,2,3,3,3,3,2,3,3,3,3,3,2,3,3,3,3,3,

%T 2,3,3,3,3,3,3,2,3,3,3,3,3,3,2,3,3,3,3,3,3,3,2,3,3,3,3,3,3,3,3,2,3,3,

%U 3,3,3,3,3,3,2,3,3,3,3,3,3,3,3,3,2,3,3,3,3,3,3,3,3,3,3,2

%N Quotient n/phi(n) for n in A007694.

%C Here phi(n) denotes Euler's totient function A000010.

%C As n increases, the proportion of 3's seems to approach 100 percent (it is 40 percent for the first 10 results; 82 percent for 100 results; 87.5 percent for 200 results while up to 200 million, for the first 235 results, is 88.51 percent). - _Zoltan Galantai_, Jul 28 2019

%C From _Bernard Schott_, Jul 30 2019: (Start)

%C According to [Ecker and Beslin], the quotients n/phi(n) when phi(n) divides n can take only 3 distinct values:

%C n/phi(n) = 1 iff n = 1,

%C n/phi(n) = 2 iff n = 2^w, w >= 1,

%C n/phi(n) = 3 iff n = 2^w * 3^u, w >= 1, u >= 1.

%C The previous comment follows because between 2^k and 2^(k+1) there are two consecutive integers for which n/phi(n) = 2, and there are floor(k*(log(2)/log(3)) integers of the form 2^b*3^c (b and c>=1) for which n/phi(n) = 3. (End)

%D Sárközy A. and Suranyi J., Number Theory Problem Book (in Hungarian), Tankonyvkiado, Budapest, 1972.

%H Jinyuan Wang, <a href="/A049237/b049237.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael W. Eckert and Scott J. Beslin, <a href="https://www.jstor.org/stable/2322340">Find all positive integers n such that phi(n) divides n</a>, AMM, Vol. 93, No 8, Oct. 86, 656-657, E 3037.

%F n/phi(n) is an integer iff n = 1 or n = 2^w*3^u for w = 1, 2, ... and u = 0, 1, 2, ...

%e For powers of 2 the quotient is 2.

%e a(95) = 124416/phi(124416) = 124416/41472 = 3.

%t Select[#/EulerPhi@ # & /@ Range[10^6], IntegerQ] (* _Michael De Vlieger_, Jul 02 2016 *)

%o (Magma) v:=[m:m in [1..150000]|m mod EulerPhi(m) eq 0];[v[k]/EulerPhi(v[k]):k in [1..#v]]; // _Marius A. Burtea_, Jul 28 2019

%o (PARI) lista(NN) = for(n=1,NN,if(n%eulerphi(n)==0,print1(n/eulerphi(n),", "))); \\ _Jinyuan Wang_, Jul 31 2019

%Y Cf. A000010, A007694, A062356.

%K nonn

%O 1,2

%A _Labos Elemer_

%E Edited by _M. F. Hasler_, Jul 02 2016