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

6, 30, 42, 60, 66, 70, 78, 90, 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, 840, 858, 870, 894, 906

Offset

1,1

Comments

Seemingly, a supersequence of A002827 (unitary perfect numbers) and a subsequence of A083207 (Zumkeller numbers).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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.

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]]; Select[Range[10^3], uZNQ] (* combined from the code by Robert G. Wilson v at A034448 and T. D. Noe at A083207 *)

Crossrefs

Cf. A002827, A034448 (sum of unitary divisors of n), A083207, A290467.

Sequence in context: A309312 A101937 A101939 * A293188 A080289 A175907

Adjacent sequences: A290463 A290464 A290465 * A290467 A290468 A290469

Keyword

nonn

Author

Ivan N. Ianakiev, Aug 03 2017

Status

approved