A160599 Composite numbers n for which n - phi(n) divides n-1.

15, 85, 255, 259, 391, 589, 1111, 3193, 4171, 4369, 12361, 17473, 21845, 25429, 28243, 47989, 52537, 65535, 65641, 68377, 83767, 91759, 100777, 120019, 144097, 167743, 186367, 268321, 286357, 291919, 316171, 327937, 335923, 346063, 353029

Offset: 1

Author

M. F. Hasler, May 23 2009

Keywords

• nonn

Comments

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).

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.

Examples

			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.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..1000
• Laurentiou Panaitopol, On some Properties Concerning the Function a(n)=n-phi(n), Bull. Greek Math. Soc., p. 71-77, Vol 45, 2001.
• Project Euler, Problem 245: resilient fractions, May 2009

Crossrefs

Cf. A160597, A160598.

Programs

Mathematica

Select[Range[400000],CompositeQ[#]&&Divisible[#-1,#-EulerPhi[#]]&] (* Harvey P. Dale, Apr 23 2019 *)

PARI

for(n=2,10^9, isprime(n) & next; (n-1)%(n-eulerphi(n)) || print1(n","))

Extensions

Offset changed from 2 to 1 by Donovan Johnson, Jan 12 2012