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

1,1

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)

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[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 *)

Crossrefs

Cf. A000010, A239886.

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

Sequence in context: A220102 A308080 A179657 * A354721 A355506 A241012

Adjacent sequences: A239884 A239885 A239886 * A239888 A239889 A239890

Keyword

nonn

Author

Olivier Gérard, Mar 29 2014

Status

approved