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

%I A055940 #21 Nov 16 2021 11:07:17
%S A055940 133,403,583,713,817,2077,2623,2923,4453,4717,5311,5773,7093,7747,
%T A055940 9313,11023,11581,11653,12877,14353,15553,19303,20803,21409,21733,
%U A055940 21971,24307,31169,35033,39283,39337,43873,46297,46357,50573,50879,53863
%N A055940 Counterbalanced numbers: Composite numbers k such that phi(k)/(sigma(k)-k) is an integer.
%C A055940 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
%H A055940 Donovan Johnson, <a href="/A055940/b055940.txt">Table of n, a(n) for n = 1..10000</a>
%H A055940 William D. Banks and Florian Luca, <a href="https://doi.org/10.1017/S0013091505001161">When the sum of aliquots divides the totient</a>, Proceedings of the Edinburgh Mathematical Society, Vol. 50, No. 3 (2007), pp. 563-569.
%H A055940 Douglas E. Iannucci, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Iannucci/ian5.html">On the Equation sigma(n) = n + phi(n)</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.2.
%e A055940 k = 133 = 7*19: phi(133)=108, sigma(133)-133 = 1+7+19 = 27, q = 4.
%t A055940 Do[s=EulerPhi[n]/(DivisorSigma[1, n]-n); If[ !PrimeQ[n]&&IntegerQ[s], Print[n]], {n, 2, 1000000}]
%t A055940 Select[Range[54000],CompositeQ[#]&&IntegerQ[EulerPhi[#]/(DivisorSigma[ 1,#]-#)]&] (* _Harvey P. Dale_, Nov 16 2021 *)
%o A055940 (PARI) is(n)=!isprime(n) && n>1 && eulerphi(n)%(sigma(n)-n)==0 \\ _Charles R Greathouse IV_, Jan 02 2014
%Y A055940 Cf. A000010, A001065, A068418, A020492, A070037.
%K A055940 nonn
%O A055940 1,1
%A A055940 _Robert G. Wilson v_, Jul 22 2000