This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A049237 #65 Jan 04 2025 07:42:41 %S A049237 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 A049237 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 A049237 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 A049237 Quotient n/phi(n) for n in A007694. %C A049237 Here phi(n) denotes Euler's totient function A000010. %C A049237 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 A049237 From _Bernard Schott_, Jul 30 2019: (Start) %C A049237 According to [Ecker and Beslin], the quotients n/phi(n) when phi(n) divides n can take only 3 distinct values: %C A049237 n/phi(n) = 1 iff n = 1, %C A049237 n/phi(n) = 2 iff n = 2^w, w >= 1, %C A049237 n/phi(n) = 3 iff n = 2^w * 3^u, w >= 1, u >= 1. %C A049237 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 A049237 Sárközy A. and Suranyi J., Number Theory Problem Book (in Hungarian), Tankonyvkiado, Budapest, 1972. %H A049237 Jinyuan Wang, <a href="/A049237/b049237.txt">Table of n, a(n) for n = 1..10000</a> %H A049237 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 A049237 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 A049237 For powers of 2 the quotient is 2. %e A049237 a(95) = 124416/phi(124416) = 124416/41472 = 3. %t A049237 Select[#/EulerPhi@ # & /@ Range[10^6], IntegerQ] (* _Michael De Vlieger_, Jul 02 2016 *) %o A049237 (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 A049237 (PARI) lista(NN) = for(n=1,NN,if(n%eulerphi(n)==0,print1(n/eulerphi(n),", "))); \\ _Jinyuan Wang_, Jul 31 2019 %Y A049237 Cf. A000010, A007694, A062356. %K A049237 nonn %O A049237 1,2 %A A049237 _Labos Elemer_ %E A049237 Edited by _M. F. Hasler_, Jul 02 2016