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%I #24 Jul 01 2022 05:25:49
%S 1,2,4,8,9,16,18,32,36,45,64,72,81,90,98,126,128,144,162,180,225,234,
%T 256,288,294,324,360,363,396,450,484,512,539,576,625,648,720,726,729,
%U 784,882,900,1008,1024,1078,1125,1152,1250,1296,1440,1452,1458,1800
%N Numbers whose divisor balance is an integer.
%H Amiram Eldar, <a href="/A023898/b023898.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Vincenzo Librandi)
%F Divisor balance of n = Sum_{d divides n} {d / phi(d)} where phi is Euler's phi function.
%e 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.
%t Select[ Range[ 2000 ], Function[ n, IntegerQ[ Plus @@ Map[ #/EulerPhi[ # ]&, Divisors[ n ] ] ] ] ]
%t dbiQ[n_]:=Module[{d=Divisors[n]},IntegerQ[Total[d/EulerPhi[d]]]]; Select[ Range[ 2000],dbiQ] (* _Harvey P. Dale_, Jul 25 2016 *)
%t q[n_] := IntegerQ[DivisorSum[n, #/EulerPhi[#] &]]; Select[Range[1800], q] (* _Amiram Eldar_, Jul 01 2022 *)
%Y Cf. A023899 (corresponding values of the divisor balance).
%Y Cf. A000010, A239886, A239887 for a version with proper divisors.
%K nonn
%O 1,2
%A _Olivier Gérard_
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