A380289 Unitary Double Zumkeller numbers: numbers whose set of unitary divisors can be partitioned into two disjoint sets with equal sums and equal cardinalities.

30, 42, 66, 78, 102, 114, 138, 150, 174, 186, 210, 222, 246, 258, 282, 294, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 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

Offset: 1

Author

Ivan N. Ianakiev, Jan 19 2025

Keywords

• nonn

Comments

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.

If y is a term, then so is x*y, where x is coprime to y.

It seems that 12k+6 is a term, where k>0 and k == 0 or 2 mod 3. Verified for k <= 191.

Examples

			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}.
		

Links

• Bhabesh Das, On unitary Zumkeller numbers, Notes on Number Theory and Discrete Mathematics, Vol. 30, No. 2 (2024), pp. 436-442.
• Eric Weisstein's World of Mathematics, Unitary Divisor Function.
• Wikipedia, Unitary divisor.

Crossrefs

Cf. A083207, A290466 (supersequence), A342398, A347063.

Programs

Mathematica

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[#]&]