A239887 Integer values of the proper divisor balance of composite numbers.

2, 4, 6, 8, 3, 8, 16, 7, 10, 24, 14, 9, 12, 32, 17, 21, 6, 14, 40, 39, 31, 9, 20, 28, 16, 48, 45, 34, 35, 18, 18, 33, 41, 56, 74, 59, 53, 19, 26, 42, 74, 50, 20, 9, 64, 73, 49, 46, 39, 32, 5, 40, 49, 32, 22, 72, 25, 109, 87, 130, 68, 21, 97, 66, 56, 137, 8, 67

Offset: 1

Author

Olivier Gérard, Mar 29 2014

Keywords

• nonn

Comments

1 and all primes have proper divisor balance 0. These values are not included in this list.

Links

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

Crossrefs

Cf. A000010, A239886.

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

Programs

Mathematica

Select[Array[Function[n, Plus @@ Map[#/EulerPhi[#] &, DeleteCases[Divisors[n], 1 | n]]], 3000], Positive[#] && IntegerQ[#] &]
Select[Table[Total[#/EulerPhi[#]&/@Most[Rest[Divisors[n]]]],{n,2,3000}],Positive[ #]&&IntegerQ[#]&] (* Harvey P. Dale, Oct 31 2020 *)
s[n_] := DivisorSum[n, #/EulerPhi[#] &, 1 < # < n &]; Select[s /@ Select[Range[10^4], CompositeQ], IntegerQ] (* 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.