A070159 Numbers k such that phi(k)/(sigma(k)-k) is an integer.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 133, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 1

Author

Labos Elemer, Apr 26 2002

Keywords

• nonn

Comments

This sequence consists of all primes p (for which the given ratio equals (p-1)/1, see A000040) and of composites listed in A055940 (see examples).

Up to 10^7, there is no element of this sequence having more than 2 prime factors. - M. F. Hasler, Dec 11 2007

Examples

			The prime p=47 is in this sequence since phi[p]/(sigma[p]-p) = p-1 is an integer, as is the case for any other prime.
The composite n=403=13*31 is in this sequence, since the ratio phi(n)/(sigma[n]-n) =360/(1+13+31)=8 is an integer.
The first few composites in this sequence are 133,403,583,713,... (A055940).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Douglas E. Iannucci, On the Equation sigma(n) = n + phi(n), Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.2.

Crossrefs

Cf. A000010, A000040, A001065, A000203, A055940, A070037, A020492, A068418, A062972.

Programs

Mathematica

Do[s=EulerPhi[n]/(DivisorSigma[1, n]-n); If[IntegerQ[s], Print[n]], {n, 2, 1000}]
Select[Range[2,300],IntegerQ[EulerPhi[#]/(DivisorSigma[1,#]-#)]&] (* Harvey P. Dale, Dec 25 2019 *)

PARI

for(n=2,999,eulerphi(n)%(sigma(n)-n) || print1(n",")) \\ M. F. Hasler, Dec 11 2007

Formula

{ a(k) } = { n in N | A000010(n)/A001065(n) is an integer }.

{ a(k) } = { A000040(k) } union { A055940(k) }.

Extensions

Edited by M. F. Hasler, Dec 11 2007