OFFSET
1,1
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
EXAMPLE
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.
MATHEMATICA
Select[Range@300, !IntegerQ@Sqrt@#&&(d=Divisors@#; MemberQ[Total/@Subsets[d, {Length@d/2}], Total@d/2])&] (* Giorgos Kalogeropoulos, Aug 15 2021 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Ivan N. Ianakiev, Aug 15 2021
EXTENSIONS
More terms from Jinyuan Wang, Aug 15 2021
STATUS
approved