A290467 Unitary half-Zumkeller numbers: numbers k whose unitary proper divisors can be partitioned into two disjoint sets whose sums are equal.

6, 12, 20, 30, 42, 56, 60, 66, 70, 72, 78, 84, 90, 102, 114, 120, 138, 150, 168, 174, 180, 186, 210, 220, 222, 240, 246, 252, 258, 272, 280, 282, 294, 318, 330, 354, 360, 364, 366, 390, 402, 420, 426, 438, 440, 462, 474, 498, 510, 520, 532, 534, 546, 560, 570, 582, 606, 618

Offset

1,1

Comments

Unitary divisors of n are divisors d such that gcd(d,n/d)=1.

Seemingly, a subsequence of A246198 (half-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 proper divisors of 12 is {1,3,4}. It can be partitioned into two disjoint subsets with equal sums of elements: {1,3} and {4}, therefore 12 is in the sequence.

Mathematica

uPropDiv[n_/; n>1]:=Block[{d=Most[Divisors[n]]}, Select[d, GCD[#, n/#]==1&]]; uhZNQ[n_]:=Module[{d=uPropDiv[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], uhZNQ] (* combined from the code by Robert G. Wilson v at A034448 and T. D. Noe at A083207 *)

Crossrefs

Cf. A083207, A246198, A290466.

Sequence in context: A080714 A116368 A343065 * A007622 A180291 A056930

Adjacent sequences: A290464 A290465 A290466 * A290468 A290469 A290470

Keyword

nonn

Author

Ivan N. Ianakiev, Aug 03 2017

Status

approved