A364053 - OEIS

%I #11 Jul 28 2024 03:49:57

%S 6,24,30,40,42,54,56,60,66,70,78,84,88,90,96,102,104,108,114,120,126,

%T 132,138,140,150,156,160,168,174,186,198,204,210,216,220,222,224,228,

%U 234,240,246,258,260,264,270,276,280,282,294,306,308,312,318,330,336,340,342,348,350,352

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

%C Intersection of A083207 and A048943.

%e 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.

%t 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]]];

%t Select[Range[1000],And[fQ[#],zQ[#]]&] (*zQ and fQ by T. D. Noe at A083207 and A048943*)

%Y Cf. A083207, A048943, A347063.

%K nonn

%O 1,1

%A _Ivan N. Ianakiev_, Jul 04 2023