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A023899 Integer values of divisor balance: Sum_{d divides k} {d / phi(d)} for numbers k in A023898. 4

%I #23 Jul 02 2022 01:56:56

%S 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,

%T 35,63,8,42,42,16,19,7,52,6,49,81,24,10,30,40,70,78,21,21,19,60,18,63,

%U 99,40,30,98,7,18,75,15,23,57,35,30,57,68,75,36,35,30,77,55,74,91,117

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

%H Amiram Eldar, <a href="/A023899/b023899.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 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.

%t Select[ Array[ Function[ n, Plus @@ Map[ #/EulerPhi[ # ]&, Divisors[ n ] ] ], 3000 ], IntegerQ ]

%t s[n_] := DivisorSum[n, #/EulerPhi[#] &]; Select[s /@ Range[4000], IntegerQ] (* _Amiram Eldar_, Jul 01 2022 *)

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

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

%K nonn

%O 1,2

%A _Olivier GĂ©rard_

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Last modified March 29 10:44 EDT 2024. Contains 371268 sequences. (Running on oeis4.)