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 A380289 #7 Feb 16 2025 08:34:07
%S A380289 30,42,66,78,102,114,138,150,174,186,210,222,246,258,282,294,318,330,
%T A380289 354,366,390,402,420,426,438,462,474,498,510,534,546,570,582,606,618,
%U A380289 630,642,654,660,678,690,714,726,750,762,770,780,786,798,822,834,858,870,894,906,910,930,942,966,978,990
%N A380289 Unitary Double Zumkeller numbers: numbers whose set of unitary divisors can be partitioned into two disjoint sets with equal sums and equal cardinalities.
%C A380289 Based on checking the first 151 terms of this sequence it seems it is a subsequence of A342398. The first number that belongs to A342398, but not to this sequence is 2394. It also seems a subsequence of Zumkeller numbers (A083207). It is not a subsequence of Sphenic numbers (A007304). For example, 150 = 2*3*5*5 does not belong to A007304.
%C A380289 If y is a term, then so is x*y, where x is coprime to y.
%C A380289 It seems that 12k+6 is a term, where k>0 and k == 0 or 2 mod 3. Verified for k <= 191.
%H A380289 Bhabesh Das, <a href="https://doi.org/10.7546/nntdm.2024.30.2.436-442">On unitary Zumkeller numbers</a>, Notes on Number Theory and Discrete Mathematics, Vol. 30, No. 2 (2024), pp. 436-442.
%H A380289 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnitaryDivisorFunction.html">Unitary Divisor Function</a>.
%H A380289 Wikipedia, <a href="https://en.wikipedia.org/wiki/Unitary_divisor">Unitary divisor</a>.
%e A380289 Let D be the set of unitary divisors of 210. D = {1,2,3,5,6,7,10,14,15,21,30,35,42,70,105,210} = {1,2,5,6,14,15,35,210}union{3,7,10,21,30,42,70,105}.
%t A380289 uDiv[n_]:=Block[{d=Divisors[n]},Select[d,GCD[#,n/#]==1&]];uZNQ[n_]:=Module[{d=uDiv[n],t,ds,x},ds=Plus@@d;If[Mod[ds,2]>0,False,t=CoefficientList[Product[1+x^i,{i,d}],x];t[[1+ds/2]]>0]];dZNQ[n_]:=Block[{div=uDiv[n]},!IntegerQ@Sqrt[n]&&MemberQ[Total/@Subsets[div,{Length@div/2}],Total@div/2]];Select[Range[1000],uZNQ[#]&&dZNQ[#]&]
%Y A380289 Cf. A083207, A290466 (supersequence), A342398, A347063.
%K A380289 nonn
%O A380289 1,1
%A A380289 _Ivan N. Ianakiev_, Jan 19 2025