A377156 Numbers k such that the set of divisors of k that are Zumkeller numbers can be partitioned into two disjoint subsets with equal sum.

120, 168, 240, 336, 432, 480, 528, 624, 660, 672, 780, 864, 924, 960, 1020, 1056, 1092, 1140, 1248, 1260, 1296, 1320, 1344, 1380, 1428, 1560, 1596, 1632, 1728, 1740, 1760, 1800, 1824, 1848, 1920, 1932, 2040, 2080, 2100, 2112, 2184, 2208, 2280, 2436, 2464, 2496, 2520, 2592

Offset: 1

Author

Ivan N. Ianakiev, Oct 18 2024

Keywords

• nonn

Comments

If k is a term, then so is 2k.

Below 6000 terms which are not Zumkeller are: 1296, 1800, 2592, 3528, 3600, 4050, 5184.

Examples

			The set D of the divisors of 120 that are Zumkeller numbers is {6,12,20,24,30,40,60,120}. D = {6,30,120} union {12,20,24,40,60}, so 120 is a term.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A083207.

Programs

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]];
zDiv[n_]:= Select[Divisors[n],zQ]; myQ[n_]:=Select[Subsets[zDiv[n]], #!={}&&EvenQ[(Plus@@zDiv[n])/2]&&Plus@@#==(Plus@@zDiv[n])/2&,1]!={};
Select[Range[336],myQ] (* zQ by T. D. Noe at A083207 *)
(* second program *)
zumQ[n_] := zumQ[n] = Module[{d = Divisors[n], sum, x}, sum = Plus @@ d; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; q[n_] := Module[{d = Select[Divisors[n], zumQ], sum, x}, sum = Plus @@ d; sum > 0 && EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; Select[Range[2600], q] (* Amiram Eldar, Oct 19 2024 *)

Extensions

More terms from Amiram Eldar, Oct 19 2024