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A332956 Symmetrical Zumkeller numbers: numbers s whose divisors can be partitioned into two disjoint sets with equal sum and whose non-divisors less than s can also be partitioned in such a way. 0

%I #14 Apr 24 2020 14:05:19

%S 12,20,24,28,40,48,56,60,80,84,88,96,104,108,112,120,132,140,156,160,

%T 168,176,180,192,204,208,216,220,224,228,240,252,260,264,272,276,280,

%U 300,304,308,312,320,336,340,348,352,360,364,368,372,380,384,396,408,416,420,432,440,444

%N Symmetrical Zumkeller numbers: numbers s whose divisors can be partitioned into two disjoint sets with equal sum and whose non-divisors less than s can also be partitioned in such a way.

%C Non-divisors of n are all the positive integers less than or equal to n that are not divisors of n.

%C Conjecture 1: Except for 6, the perfect numbers (A000396) are in the sequence.

%C Conjecture 2: At least one out of every twenty consecutive natural numbers is such a number.

%C Conjecture 3: At least one out of every four consecutive Zumkeller numbers is such a number.

%e The divisors of 12 are D = {1,2,3,4,6,12} = {1,3,4,6} Union {2,12}. The non-divisors of 12 are N = {5,7,8,9,10,11} = {5,9,11} Union {7,8,10}. Therefore, 12 is in the sequence.

%t nondivisors[n_]:=Complement[Range[n],Divisors[n]];

%t azQ[n_]:=Module[{d=nondivisors[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]];

%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]];

%t Select[Range[500],zQ[#]&&azQ[#]&] (* code by _T. D. Noe_ at A083207 is used *)

%Y Cf. A000396, A083207 (supersequence).

%K nonn

%O 1,1

%A _Ivan N. Ianakiev_, Mar 04 2020

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