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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
12, 20, 24, 28, 40, 48, 56, 60, 80, 84, 88, 96, 104, 108, 112, 120, 132, 140, 156, 160, 168, 176, 180, 192, 204, 208, 216, 220, 224, 228, 240, 252, 260, 264, 272, 276, 280, 300, 304, 308, 312, 320, 336, 340, 348, 352, 360, 364, 368, 372, 380, 384, 396, 408, 416, 420, 432, 440, 444
OFFSET
1,1
COMMENTS
Non-divisors of n are all the positive integers less than or equal to n that are not divisors of n.
Conjecture 1: Except for 6, the perfect numbers (A000396) are in the sequence.
Conjecture 2: At least one out of every twenty consecutive natural numbers is such a number.
Conjecture 3: At least one out of every four consecutive Zumkeller numbers is such a number.
EXAMPLE
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.
MATHEMATICA
nondivisors[n_]:=Complement[Range[n], Divisors[n]];
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]];
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]];
Select[Range[500], zQ[#]&&azQ[#]&] (* code by T. D. Noe at A083207 is used *)
CROSSREFS
Cf. A000396, A083207 (supersequence), A279064 (supersequence).
Sequence in context: A360253 A389144 A097320 * A386417 A204825 A111592
KEYWORD
nonn
AUTHOR
Ivan N. Ianakiev, Mar 04 2020
STATUS
approved