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

1,1

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)

Formula

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

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 *)

Crossrefs

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

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

Sequence in context: A312778 A312779 A312780 * A312781 A312782 A312783

Adjacent sequences: A239883 A239884 A239885 * A239887 A239888 A239889

Keyword

nonn

Author

Olivier Gérard, Mar 29 2014

Status

approved