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

1,1

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.

Links

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

Example

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.

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 *)
(* Alternative: *)
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 *)

Crossrefs

Cf. A083207.

Sequence in context: A030634 A396323 A272594 * A189975 A232461 A090782

Adjacent sequences: A377153 A377154 A377155 * A377157 A377158 A377159

Keyword

nonn

Author

Ivan N. Ianakiev, Oct 18 2024

Extensions

More terms from Amiram Eldar, Oct 19 2024

Status

approved