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A347063
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Double Zumkeller numbers: numbers whose set of divisors can be partitioned into two disjoint subsets with equal sums and equal cardinalities.
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2
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24, 30, 42, 48, 54, 60, 66, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 140, 150, 156, 160, 168, 174, 180, 186, 192, 198, 204, 210, 216, 220, 222, 224, 228, 240, 246, 252, 258, 260, 264, 270, 276, 280, 282, 300, 306, 308, 312, 318, 320, 330, 336, 340, 342
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OFFSET
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1,1
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COMMENTS
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If x is a Zumkeller number, then so is 2x. Conjecturally, if y is a term of this sequence, then so is 2y.
If y is a term of this sequence, then so is p*y, where p is a prime that is coprime to y. Proof: Let D = {d_1,d_2,...,d_k} be the set of divisors of y. Let E be the set of divisors of p*y. Except for the divisors of y E contains their products with p. In other words, E = {d_1,d_2,...,d_2k}, meaning that the cardinality of E is twice the cardinality of D. Those additional divisors are F = {p*d_1,p*d_2,...,p*d_k}. Since D can be partitioned into two disjoint subsets with equal sums and cardinalities by definition, this must be true about F and also about E = D union F. QED. - Ivan N. Ianakiev, Nov 20 2021
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LINKS
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EXAMPLE
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The set of divisors of 24 is D = {1,2,3,4,6,8,12,24}. D = {1,2,3,24} union {4,6,8,12}, so 24 is in the sequence.
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MATHEMATICA
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Select[Range@300, !IntegerQ@Sqrt@#&&(d=Divisors@#;
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CROSSREFS
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Subsequence of A083207 (Zumkeller numbers).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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