A160598 -id:A160598 - cp's OEIS frontend

A160597 Denominator of coresilience C(n) = (n - phi(n))/(n-1).

1, 2, 3, 4, 5, 6, 7, 8, 3, 10, 11, 12, 13, 2, 15, 16, 17, 18, 19, 20, 7, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 11, 34, 35, 36, 37, 38, 13, 40, 41, 42, 43, 44, 15, 46, 47, 48, 49, 50, 51, 52, 53, 18, 55, 8, 19, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 3, 70, 71, 72, 73, 74

Offset: 2

M. F. Hasler, May 23 2009

Obviously C(p) = 1/(p-1) for all primes p.

			a(10)=3 since for n=10, we have (n - phi(n))/(n-1) = (10-4)/9 = 2/3.

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

Cf. A160598.

[Denominator((n-EulerPhi(n))/(n-1)): n in [2..80]]; // Vincenzo Librandi, Dec 27 2016

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

Denominator[Table[(n - EulerPhi[n])/(n - 1), {n, 2, 20}]] (* G. C. Greubel, Dec 26 2016 *)

A160597(n)=denominator((n-eulerphi(n))/(n-1))

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

Original entry on oeis.org

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

M. F. Hasler, May 23 2009

nonn

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.

			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.

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

Cf. A160597, A160598.

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

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

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

cp's OEIS Frontend

A160597 Denominator of coresilience C(n) = (n - phi(n))/(n-1).

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