A347063 Double Zumkeller numbers: numbers whose set of divisors can be partitioned into two disjoint subsets with equal sums and equal cardinalities.

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

Offset: 1

Author

Ivan N. Ianakiev, Aug 15 2021

Keywords

• nonn

Comments

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

It seems that for k>=1 all numbers of the form 18k+12 are terms. Verified for k in [1, 45]. - Ivan N. Ianakiev, Oct 01 2024

Examples

			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.
		

Crossrefs

Subsequence of A083207 (Zumkeller numbers).

Programs

Mathematica

Select[Range@300,!IntegerQ@Sqrt@#&&(d=Divisors@#; MemberQ[Total/@Subsets[d,{Length@d/2}],Total@d/2])&] (* Giorgos Kalogeropoulos, Aug 15 2021 *)

Extensions

More terms from Jinyuan Wang, Aug 15 2021