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A290466 Unitary Zumkeller numbers: numbers k whose unitary divisors can be partitioned into two disjoint subsets whose sums are both usigma(k)/2.

%I A290466 #21 Feb 16 2025 08:33:49
%S A290466 6,30,42,60,66,70,78,90,102,114,138,150,174,186,210,222,246,258,282,
%T A290466 294,318,330,354,366,390,402,420,426,438,462,474,498,510,534,546,570,
%U A290466 582,606,618,630,642,654,660,678,690,714,726,750,762,770,780,786,798,822,834,840,858,870,894,906
%N A290466 Unitary Zumkeller numbers: numbers k whose unitary divisors can be partitioned into two disjoint subsets whose sums are both usigma(k)/2.
%C A290466 Seemingly, a supersequence of A002827 (unitary perfect numbers) and a subsequence of A083207 (Zumkeller numbers).
%H A290466 Amiram Eldar, <a href="/A290466/b290466.txt">Table of n, a(n) for n = 1..10000</a>
%H A290466 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 A290466 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnitaryDivisorFunction.html">Unitary Divisor Function</a>.
%H A290466 Wikipedia, <a href="https://en.wikipedia.org/wiki/Unitary_divisor">Unitary divisor</a>.
%e A290466 The set of unitary divisors of 30 is {1,2,3,5,6,10,15,30}. It can be partitioned into two disjoint subsets with equal sums of elements: {5,6,10,15} and {1,2,3,30}, therefore 30 is in the sequence.
%t A290466 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]];Select[Range[10^3],uZNQ] (* combined from the code by _Robert G. Wilson v_ at A034448 and _T. D. Noe_ at A083207 *)
%Y A290466 Cf. A002827, A034448 (sum of unitary divisors of n), A083207, A290467.
%K A290466 nonn
%O A290466 1,1
%A A290466 _Ivan N. Ianakiev_, Aug 03 2017