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%I #21 Apr 23 2019 18:15:29
%S 15,85,255,259,391,589,1111,3193,4171,4369,12361,17473,21845,25429,
%T 28243,47989,52537,65535,65641,68377,83767,91759,100777,120019,144097,
%U 167743,186367,268321,286357,291919,316171,327937,335923,346063,353029
%N Composite numbers n for which n - phi(n) divides n-1.
%C Obviously, C(p) = (p-phi(p))/(p-1) = 1/(p-1), i.e., A160598(p)=1, for all primes p. This sequence lists composite numbers for which C(n) has denominator 1, i.e., n-1 is a multiple of n - phi(n).
%C The sequence contains numbers F(k)*F(k+1)*...*F(k+d), if the factors are successive Fermat primes F(k)=2^(2^k)+1.
%H Donovan Johnson, <a href="/A160599/b160599.txt">Table of n, a(n) for n = 1..1000</a>
%H Laurentiou Panaitopol, <a href="http://www.hms.gr/apothema/?s=sa&i=733">On some Properties Concerning the Function a(n)=n-phi(n)</a>, Bull. Greek Math. Soc., p. 71-77, Vol 45, 2001.
%H Project Euler, <a href="http://projecteuler.net/index.php?section=problems&id=245">Problem 245: resilient fractions</a>, May 2009
%e a(1)=15 is in the sequence, because for n=15, we have (n - phi(n))/(n-1) = (15-8)/14 = 1/2; apart from the primes, this is the smallest number n such that C(n) is a unit fraction.
%t Select[Range[400000],CompositeQ[#]&&Divisible[#-1,#-EulerPhi[#]]&] (* _Harvey P. Dale_, Apr 23 2019 *)
%o (PARI) for(n=2,10^9, isprime(n) & next; (n-1)%(n-eulerphi(n)) | print1(n","))
%Y Cf. A160597, A160598.
%K nonn
%O 1,1
%A _M. F. Hasler_, May 23 2009
%E Offset changed from 2 to 1 by _Donovan Johnson_, Jan 12 2012
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