A239887 -id:A239887 - 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.

A023899 Integer values of divisor balance: Sum_{d divides k} {d / phi(d)} for numbers k in A023898.

Original entry on oeis.org

1, 3, 5, 7, 4, 9, 12, 11, 20, 9, 13, 28, 7, 27, 10, 26, 15, 36, 21, 45, 14, 25, 17, 44, 25, 35, 63, 8, 42, 42, 16, 19, 7, 52, 6, 49, 81, 24, 10, 30, 40, 70, 78, 21, 21, 19, 60, 18, 63, 99, 40, 30, 98, 7, 18, 75, 15, 23, 57, 35, 30, 57, 68, 75, 36, 35, 30, 77, 55, 74, 91, 117

Offset: 1

Olivier Gérard

nonn

			a(9) = 20 because the 9th integer having an integer divisor balance is 36 : its divisors are 1,2,3,4,6,9,12,18 and 36, giving the fractions (1, 2, 3/2, 2, 3, 3/2, 3, 3, 3) which sum to 20.

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

cf. A023898 (integers for which the divisor balance is an integer).

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

Select[ Array[ Function[ n, Plus @@ Map[ #/EulerPhi[ # ]&, Divisors[ n ] ] ], 3000 ], IntegerQ ]
s[n_] := DivisorSum[n, #/EulerPhi[#] &]; Select[s /@ Range[4000], IntegerQ] (* 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.

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A023898 Numbers whose divisor balance is an integer.

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A023899 Integer values of divisor balance: Sum_{d divides k} {d / phi(d)} for numbers k in A023898.

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A239886 Composite numbers whose proper divisor balance is an integer.

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