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

Author

Ivan N. Ianakiev, Aug 03 2017

Keywords

• nonn

Comments

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

Seemingly, a subsequence of A246198 (half-Zumkeller numbers).

The conjecture above is false, since 72, 3600 and 19600 do not belong to A246198. - Ivan N. Ianakiev, Jan 08 2025

Examples

			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.
		

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.

Crossrefs

Cf. A083207, A246198, A290466.

Programs

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 *)