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

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

Author

Olivier Gérard

Keywords

• nonn

Examples

			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.
		

Links

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

Crossrefs

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

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

Programs

Mathematica

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 *)

Formula

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