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

1,1

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.

Links

Table of n, a(n) for n=1..61.

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.

Example

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}.

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[#]&]

Crossrefs

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

Sequence in context: A160352 A291446 A342398 * A226104 A091455 A367481

Adjacent sequences: A380286 A380287 A380288 * A380290 A380291 A380292

Keyword

nonn

Author

Ivan N. Ianakiev, Jan 19 2025

Status

approved