A364053 Zumkeller numbers whose divisors can be partitioned into two disjoint sets with equal products.

6, 24, 30, 40, 42, 54, 56, 60, 66, 70, 78, 84, 88, 90, 96, 102, 104, 108, 114, 120, 126, 132, 138, 140, 150, 156, 160, 168, 174, 186, 198, 204, 210, 216, 220, 222, 224, 228, 234, 240, 246, 258, 260, 264, 270, 276, 280, 282, 294, 306, 308, 312, 318, 330, 336, 340, 342, 348, 350, 352

Offset

1,1

Comments

Intersection of A083207 and A048943.

Links

Table of n, a(n) for n=1..60.

Example

The divisors of 24 are {1,2,3,4,6,8,12,24}. They can be partitioned into two disjoint sets with equal sums, namely {4,6,8,12} and {1,2,3,24}, and two disjoint sets with equal products, namely {1,2,12,24} and {3,4,6,8}. So, 24 is a term and also a term of A347063.

Mathematica

zQ[n_]:=Module[{d=Divisors[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]]; fQ[n_]:=IntegerQ[Sqrt[Times@@Divisors[n]]];
Select[Range[1000], And[fQ[#], zQ[#]]&] (*zQ and fQ by T. D. Noe at A083207 and A048943*)

Crossrefs

Cf. A083207, A048493, A347063.

Sequence in context: A335215 A292985 A335197 * A306983 A234648 A110926

Adjacent sequences: A364050 A364051 A364052 * A364054 A364055 A364056

Keyword

nonn

Author

Ivan N. Ianakiev, Jul 04 2023

Status

approved