A239886 Composite numbers whose proper divisor balance is an integer.

4, 8, 16, 18, 27, 32, 36, 50, 64, 72, 100, 105, 128, 144, 162, 200, 243, 256, 288, 300, 324, 375, 392, 400, 512, 576, 648, 700, 800, 850, 1024, 1100, 1134, 1152, 1200, 1296, 1350, 1352, 1458, 1600, 1620, 1650, 2048, 2187, 2304, 2592, 2850, 2916, 3078, 3100, 3125

Offset: 1

Author

Olivier Gérard, Mar 29 2014

Keywords

• nonn

Comments

This list excludes 1 and prime numbers, which by definition have no proper divisors.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..267 from Vincenzo Librandi)

Crossrefs

Cf. A239887 (corresponding values of the proper divisor balance).

Cf. A000010, A023898, A023899 for the versions with all divisors included.

Programs

Mathematica

Select[Range[2000], Function[ n, ! PrimeQ[n] &&
   IntegerQ[Plus @@ Map[#/EulerPhi[#] &, DeleteCases[Divisors[n], 1 | n]]]]]
q[n_] := CompositeQ[n] && IntegerQ[DivisorSum[n, #/EulerPhi[#] &, 1 < # < n &]]; Select[Range[4000], q] (* Amiram Eldar, Jul 01 2022 *)

Formula

Proper divisor balance of n = Sum_{1 < d < n and divides n} {d / phi(d)} where phi is Euler's phi function.