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A160595 Numerator of resilience R(n) = phi(n)/(n-1), with a(1) = 1 by convention.

%I A160595 #48 Sep 08 2022 08:45:45
%S A160595 1,1,1,2,1,2,1,4,3,4,1,4,1,6,4,8,1,6,1,8,3,10,1,8,5,12,9,4,1,8,1,16,5,
%T A160595 16,12,12,1,18,12,16,1,12,1,20,6,22,1,16,7,20,16,8,1,18,20,24,9,28,1,
%U A160595 16,1,30,18,32,3,4,1,32,11,8,1,24,1,36,20,12,15,24,1,32,27,40,1,24,16,42,28
%N A160595 Numerator of resilience R(n) = phi(n)/(n-1), with a(1) = 1 by convention.
%C A160595 The resilience of a denominator, R(d), is the ratio of proper fractions n/d, 0 < n < d, that are resilient, i.e., such that gcd(n, d) = 1. Obviously this is the case for phi(d) proper fractions among the d - 1 possible ones.
%C A160595 a(n) = 1 if n is prime. It is unknown whether there exist composite n with a(n) = 1 (see Wikipedia link). - _Robert Israel_, Dec 26 2016
%C A160595 Conjecture: a(n) > 2 for every composite n > 6. Slightly stronger than the Lehmer's totient conjecture (1932). - _Thomas Ordowski_, Mar 13 2019
%H A160595 Antti Karttunen, <a href="/A160595/b160595.txt">Table of n, a(n) for n = 1..65537</a> (terms 2..10000 from Robert Israel)
%H A160595 Project Euler, <a href="http://projecteuler.net/index.php?section=problems&amp;id=245">Problem 245: resilient fractions</a>, May 2009
%H A160595 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lehmer&#39;s_totient_problem">Lehmer's totient problem</a>.
%F A160595 a(n) = phi(n)/gcd(phi(n),n-1) = A000010(n) / A049559(n) = A247074(n) * A318829(n). - _Antti Karttunen_, Sep 09 2018
%e A160595 a(9) = 3 since for the denominator d = 9, among the 8 proper fractions n/9 (n = 1, ..., 8), six cannot be canceled down by a common factor (namely 1/9, 2/9, 4/9, 5/9, 7/9, 8/9), thus R(9) = 6/8 = 3/4.
%p A160595 seq(numer(numtheory:-phi(n)/(n-1)),n=2..100); # _Robert Israel_, Dec 26 2016
%t A160595 Numerator[Table[EulerPhi[n]/(n - 1), {n, 2, 87}]] (* _Alonso del Arte_, Sep 19 2011 *)
%o A160595 (PARI) A160595(n) = if(1==n,n,numerator(eulerphi(n)/(n-1)));
%o A160595 (Magma) [Numerator(EulerPhi(n)/(n-1)): n in [2..90]]; // _Vincenzo Librandi_, Jan 02 2017
%Y A160595 Cf. A000010, A049559, A247074, A318829.
%K A160595 nonn,look
%O A160595 1,4
%A A160595 _M. F. Hasler_, May 23 2009
%E A160595 Term a(1) = 1 prepended by _Antti Karttunen_, Sep 09 2018