This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A347063 #36 Oct 05 2024 11:00:11
%S A347063 24,30,42,48,54,60,66,78,84,90,96,102,108,114,120,126,132,138,140,150,
%T A347063 156,160,168,174,180,186,192,198,204,210,216,220,222,224,228,240,246,
%U A347063 252,258,260,264,270,276,280,282,300,306,308,312,318,320,330,336,340,342
%N A347063 Double Zumkeller numbers: numbers whose set of divisors can be partitioned into two disjoint subsets with equal sums and equal cardinalities.
%C A347063 If x is a Zumkeller number, then so is 2x. Conjecturally, if y is a term of this sequence, then so is 2y.
%C A347063 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
%C A347063 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
%e A347063 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.
%t A347063 Select[Range@300,!IntegerQ@Sqrt@#&&(d=Divisors@#; MemberQ[Total/@Subsets[d,{Length@d/2}],Total@d/2])&] (* _Giorgos Kalogeropoulos_, Aug 15 2021 *)
%Y A347063 Subsequence of A083207 (Zumkeller numbers).
%K A347063 nonn
%O A347063 1,1
%A A347063 _Ivan N. Ianakiev_, Aug 15 2021
%E A347063 More terms from _Jinyuan Wang_, Aug 15 2021