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A285510
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Numbers k such that the average of the squarefree divisors of k is an integer.
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3
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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
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OFFSET
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1,2
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COMMENTS
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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).
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)
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LINKS
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FORMULA
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a(n) ~ n (conjecture).
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EXAMPLE
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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 integer.
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MAPLE
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filter:= n -> n::odd or has(numtheory:-factorset(n) mod 4, 3):
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A001221, A001615, A002145, A003601, A005117, A007947, A023886, A034444, A048250, A072437, A078174, A103826, A206778.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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