A023899 -id:A023899 - cp's OEIS frontend

A023898 Numbers whose divisor balance is an integer.

1, 2, 4, 8, 9, 16, 18, 32, 36, 45, 64, 72, 81, 90, 98, 126, 128, 144, 162, 180, 225, 234, 256, 288, 294, 324, 360, 363, 396, 450, 484, 512, 539, 576, 625, 648, 720, 726, 729, 784, 882, 900, 1008, 1024, 1078, 1125, 1152, 1250, 1296, 1440, 1452, 1458, 1800

Offset: 1

Olivier Gérard

nonn

			45 is in this list because its divisors are 1,3,5,9,15 and 45, the corresponding fractions are : (1, 3/2, 5/4, 3/2, 15/8, 15/8) and their sum is 9, which is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Cf. A023899 (corresponding values of the divisor balance).

Cf. A000010, A239886, A239887 for a version with proper divisors.

Select[ Range[ 2000 ], Function[ n, IntegerQ[ Plus @@ Map[ #/EulerPhi[ # ]&, Divisors[ n ] ] ] ] ]
dbiQ[n_]:=Module[{d=Divisors[n]},IntegerQ[Total[d/EulerPhi[d]]]]; Select[ Range[ 2000],dbiQ] (* Harvey P. Dale, Jul 25 2016 *)
q[n_] := IntegerQ[DivisorSum[n, #/EulerPhi[#] &]]; Select[Range[1800], q] (* Amiram Eldar, Jul 01 2022 *)

Divisor balance of n = Sum_{d divides n} {d / phi(d)} where phi is Euler's phi function.

A239886 Composite numbers whose proper divisor balance is an integer.

Original entry on oeis.org

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

Olivier Gérard, Mar 29 2014

nonn

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

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

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

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

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 *)

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

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

Original entry on oeis.org

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

Olivier Gérard, Mar 29 2014

nonn

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

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

Cf. A000010, A239886.

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

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 *)

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

cp's OEIS Frontend

A023898 Numbers whose divisor balance is an integer.

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A239886 Composite numbers whose proper divisor balance is an integer.

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

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Mathematica

Formula