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

%I A376862 #11 Feb 16 2025 08:34:07
%S A376862 30,42,60,66,78,90,102,114,138,150,174,186,210,222,246,258,282,318,
%T A376862 330,354,366,390,402,420,426,438,462,474,498,510,534,546,570,582,606,
%U A376862 618,630,642,654,660,678,690
%N A376862 Unitary Zumkeller numbers whose divisors can be partitioned into two disjoint subsets with equal sums and cardinalities.
%C A376862 A unitary divisor of n is a divisor d such that gcd(d,n/d)=1.
%C A376862 This sequence is an intersection of A290466 and A347063 and seemingly a subsequence of A293188.
%C A376862 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.
%C A376862 It seems that for k >= 0 all numbers of the form 30 + 36k are terms.
%H A376862 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnitaryDivisorFunction.html">Unitary Divisor Function</a>
%H A376862 Wikipedia, <a href="http://en.wikipedia.org/wiki/Unitary_divisor">Unitary divisor</a>
%e A376862 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.
%t A376862 uzn=Cases[Import["https://oeis.org/A290466/b290466.txt","Table"],{_,_}][[All,2]];
%t A376862 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 *)
%Y A376862 Cf. A083207, A290466, A347063.
%K A376862 nonn,more
%O A376862 1,1
%A A376862 _Ivan N. Ianakiev_, Oct 07 2024