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

%I A285510 #33 Jan 11 2025 13:47:10
%S A285510 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,
%T A285510 35,36,37,38,39,41,42,43,44,45,46,47,48,49,51,53,54,55,56,57,59,60,61,
%U A285510 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
%N A285510 Numbers k such that the average of the squarefree divisors of k is an integer.
%C A285510 Numbers n such that A034444(n)|A048250(n).
%C A285510 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).
%C A285510 From _Robert Israel_, Apr 24 2017: (Start)
%C A285510 All odd numbers are in the sequence.
%C A285510 A positive even number is in the sequence if and only if at least one of its prime factors is in A002145.
%C A285510 Thus this is the complement of 2*A072437 in the positive numbers.
%C A285510 (End)
%H A285510 Robert Israel, <a href="/A285510/b285510.txt">Table of n, a(n) for n = 1..10000</a>
%H A285510 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.
%F A285510 a(n) ~ n (conjecture).
%F A285510 Conjecture is true, since A072437 has density 0. - _Robert Israel_, Apr 24 2017
%e A285510 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.
%p A285510 filter:= n -> n::odd or has(numtheory:-factorset(n) mod 4, 3):
%p A285510 select(filter, [$1..1000]); # _Robert Israel_, Apr 24 2017
%t A285510 Select[Range[100], IntegerQ[Total[Select[Divisors[#], SquareFreeQ]] / 2^PrimeNu[#]] &]
%t A285510 Select[Range[110],IntegerQ[Mean[Select[Divisors[#],SquareFreeQ]]]&] (* _Harvey P. Dale_, Apr 11 2018 *)
%t A285510 Select[Range[100], IntegerQ[Times @@ ((1 + FactorInteger[#][[;; , 1]])/2)] &] (* _Amiram Eldar_, Jul 01 2022 *)
%Y A285510 Cf. A001221, A001615, A002145, A003601, A005117, A007947, A023886, A034444, A048250, A072437, A078174, A103826, A206778.
%K A285510 nonn
%O A285510 1,2
%A A285510 _Ilya Gutkovskiy_, Apr 20 2017