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

1,2

Links

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

Formula

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

Example

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.

Mathematica

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 *)

Crossrefs

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

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

Sequence in context: A114400 A341742 A330710 * A125853 A080025 A152111

Adjacent sequences: A023895 A023896 A023897 * A023899 A023900 A023901

Keyword

nonn

Author

Olivier Gérard

Status

approved