A055940 Counterbalanced numbers: Composite numbers k such that phi(k)/(sigma(k)-k) is an integer.

133, 403, 583, 713, 817, 2077, 2623, 2923, 4453, 4717, 5311, 5773, 7093, 7747, 9313, 11023, 11581, 11653, 12877, 14353, 15553, 19303, 20803, 21409, 21733, 21971, 24307, 31169, 35033, 39283, 39337, 43873, 46297, 46357, 50573, 50879, 53863

Offset: 1

Author

Robert G. Wilson v, Jul 22 2000

Keywords

• nonn

Comments

Banks and Luca (2007) showed that the number of terms <= x, N(x) <= x * exp(-((1/3)*(log(8))^(1/3) + o(1))*(log(x))^(1/3)*(log(log(x)))^(1/3)) as x -> infinity, and that under Dickson's conjecture this sequence is infinite, since for each positive integer m, if p = 5m + 1 and q = 20m + 13 are primes, then p*q is a term. - Amiram Eldar, Apr 13 2020

Examples

			k = 133 = 7*19: phi(133)=108, sigma(133)-133 = 1+7+19 = 27, q = 4.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..10000
• William D. Banks and Florian Luca, When the sum of aliquots divides the totient, Proceedings of the Edinburgh Mathematical Society, Vol. 50, No. 3 (2007), pp. 563-569.
• 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, A001065, A068418, A020492, A070037.

Programs

Mathematica

Do[s=EulerPhi[n]/(DivisorSigma[1, n]-n); If[ !PrimeQ[n]&&IntegerQ[s], Print[n]], {n, 2, 1000000}]
Select[Range[54000],CompositeQ[#]&&IntegerQ[EulerPhi[#]/(DivisorSigma[ 1,#]-#)]&] (* Harvey P. Dale, Nov 16 2021 *)

PARI

is(n)=!isprime(n) && n>1 && eulerphi(n)%(sigma(n)-n)==0 \\ Charles R Greathouse IV, Jan 02 2014