A160596 Denominator of resilience R(n) = phi(n)/(n-1).

1, 1, 3, 1, 5, 1, 7, 4, 9, 1, 11, 1, 13, 7, 15, 1, 17, 1, 19, 5, 21, 1, 23, 6, 25, 13, 9, 1, 29, 1, 31, 8, 33, 17, 35, 1, 37, 19, 39, 1, 41, 1, 43, 11, 45, 1, 47, 8, 49, 25, 17, 1, 53, 27, 55, 14, 57, 1, 59, 1, 61, 31, 63, 4, 13, 1, 67, 17, 23, 1, 71, 1, 73, 37, 25, 19, 77, 1, 79, 40, 81, 1

Offset: 2

Author

M. F. Hasler, May 23 2009

Keywords

• nonn
• look

Comments

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.

a(n) = 1 if and only if n is prime. - Robert Israel, Dec 26 2016

Examples

			a(9)=4 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.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..10000
• Project Euler, Problem 245: resilient fractions, May 2009

Programs

Magma

[Denominator(EulerPhi(n)/(n-1)): n in [2..80]]; // Vincenzo Librandi, Jan 02 2017

Maple

seq(denom(numtheory:-phi(n)/(n-1)),n=2..100); # Robert Israel, Dec 26 2016

Mathematica

Denominator[Table[EulerPhi[n]/(n-1),{n,2,90}]] (* Harvey P. Dale, Apr 18 2012 *)

PARI

A160496(n)=denominator(eulerphi(n)/(n-1))