login
A376862
Unitary Zumkeller numbers whose divisors can be partitioned into two disjoint subsets with equal sums and cardinalities.
0
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
1,1
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.
LINKS
Eric Weisstein's World of Mathematics, Unitary Divisor Function
Wikipedia, Unitary divisor
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Ivan N. Ianakiev, Oct 07 2024
STATUS
approved