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]];
CROSSREFS
KEYWORD
nonn
AUTHOR
Ivan N. Ianakiev, Mar 04 2020
STATUS
approved