A376862 Unitary Zumkeller numbers whose divisors can be partitioned into two disjoint subsets with equal sums and cardinalities.

30, 42, 60, 66, 78, 90, 102, 114, 138, 150, 174, 186, 210, 222, 246, 258, 282, 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

Offset: 1

Author

Ivan N. Ianakiev, Oct 07 2024

Keywords

• nonn
• more

Comments

A unitary divisor of n is a divisor d such that gcd(d,n/d)=1.

This sequence is an intersection of A290466 and A347063 and seemingly a subsequence of A293188.

From the facts: a) for n>2 every primorial(n), i.e. A002110(n), is a Zumkeller number, b) a(1) = 30 = 2*3*5 is primorial(3), c) if n is squarefree, than sigma(n) = usigma(n), d) the number of unitary divisors of n is 2^k, where k is the number of distinct prime factors of n, and e) p*y belongs to A347063, where p is a prime coprime to y and y belongs to A347063, it follows that the present sequence is infinite, since for m >= 3 primorial(m) is a term.

It seems that for k >= 0 all numbers of the form 30 + 36k are terms.

Examples

			The set of divisors of 90 is {1,2,3,5,6,9,10,15,18,30,45,90}, which is a union of the sets {1,2,3,6,15,90} and {5,9,10,18,30,45}, which have equal sums (117) and cardinalities (6). So, 90 is a term.
		

Links

• Eric Weisstein's World of Mathematics, Unitary Divisor Function
• Wikipedia, Unitary divisor

Crossrefs

Cf. A083207, A290466, A347063.

Programs

Mathematica

uzn=Cases[Import["https://oeis.org/A290466/b290466.txt","Table"],{,}][[All,2]];
dzn=Select[Range@700,!IntegerQ@Sqrt@#&&(d=Divisors@#; MemberQ[Total/@Subsets[d,{Length@d/2}],Total@d/2])&]; Intersection[uzn,dzn] (* Thanks to Giorgos Kalogeropoulos at A347063 *)