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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.

%I A377156 #22 Oct 19 2024 22:12:14
%S A377156 120,168,240,336,432,480,528,624,660,672,780,864,924,960,1020,1056,
%T A377156 1092,1140,1248,1260,1296,1320,1344,1380,1428,1560,1596,1632,1728,
%U A377156 1740,1760,1800,1824,1848,1920,1932,2040,2080,2100,2112,2184,2208,2280,2436,2464,2496,2520,2592
%N 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.
%C A377156 If k is a term, then so is 2k.
%C A377156 Below 6000 terms which are not Zumkeller are: 1296, 1800, 2592, 3528, 3600, 4050, 5184.
%H A377156 Amiram Eldar, <a href="/A377156/b377156.txt">Table of n, a(n) for n = 1..1000</a>
%e A377156 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.
%t A377156 zQ[n_]:=Module[{d=Divisors[n],t,ds,x},ds=Plus@@d;If[Mod[ds,2]>0,False,
%t A377156 t=CoefficientList[Product[1+x^i,{i,d}],x];t[[1+ds/2]]>0]];
%t A377156 zDiv[n_]:= Select[Divisors[n],zQ]; myQ[n_]:=Select[Subsets[zDiv[n]], #!={}&&EvenQ[(Plus@@zDiv[n])/2]&&Plus@@#==(Plus@@zDiv[n])/2&,1]!={};
%t A377156 Select[Range[336],myQ] (* zQ by _T. D. Noe_ at A083207 *)
%t A377156 (* second program *)
%t A377156 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 *)
%Y A377156 Cf. A083207.
%K A377156 nonn
%O A377156 1,1
%A A377156 _Ivan N. Ianakiev_, Oct 18 2024
%E A377156 More terms from _Amiram Eldar_, Oct 19 2024