A285510 Numbers k such that the average of the squarefree divisors of k is an integer.

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 81, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101

Offset: 1

Author

Ilya Gutkovskiy, Apr 20 2017

Keywords

• nonn

Comments

Numbers n such that A034444(n)|A048250(n).

Numbers n such that 2^omega(n)|psi(rad(n)), where omega() is the number of distinct prime divisors (A001221), psi() is the Dedekind psi function (A001615) and rad() is the squarefree kernel (A007947).

From Robert Israel, Apr 24 2017: (Start)

All odd numbers are in the sequence.

A positive even number is in the sequence if and only if at least one of its prime factors is in A002145.

Thus this is the complement of 2*A072437 in the positive numbers.

(End)

Examples

			44 is in the sequence because 44 has 6 divisors {1, 2, 4, 11, 22, 44} among which 4 are squarefree {1, 2, 11, 22} and (1 + 2 + 11 + 22)/4 = 9 is an integer.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to sums of divisors.

Crossrefs

Cf. A001221, A001615, A002145, A003601, A005117, A007947, A023886, A034444, A048250, A072437, A078174, A103826, A206778.

Programs

Maple

filter:= n -> n::odd or has(numtheory:-factorset(n) mod 4, 3):
select(filter, [$1..1000]); # Robert Israel, Apr 24 2017

Mathematica

Select[Range[100], IntegerQ[Total[Select[Divisors[#], SquareFreeQ]] / 2^PrimeNu[#]] &]
Select[Range[110],IntegerQ[Mean[Select[Divisors[#],SquareFreeQ]]]&] (* Harvey P. Dale, Apr 11 2018 *)
Select[Range[100], IntegerQ[Times @@ ((1 + FactorInteger[#][[;; , 1]])/2)] &] (* Amiram Eldar, Jul 01 2022 *)

Formula

a(n) ~ n (conjecture).

Conjecture is true, since A072437 has density 0. - Robert Israel, Apr 24 2017