Colin Mallows's problems

[Max]

Ralf Stephan wrote:
> Nothing wow. Found an error, it should be:
> 
> %S A000001 1,2,2,4,2,6,2,8,4,6,2,17,2,6,6,16,2,18,2,17
> %N A000001 Number of polynomials with coefficients in {0,1} that divide x^n-1.
> (unchanged)

I've computed more terms:
  1 2 2 4 2 6 2 8 4 6 2 17 2 6 6 16 2 18 2 17 6 6 2 48 4 6 8 17 2 36 2 32 6 6 6 69 2 6 6 47 2 36 2 17 17 6 2 136 4 18 6 17 2 54 6 47 6 6 2 176 2 6 17 64 6 36 2 17 6 36 2 257 2 6 18 17 6 36 2 131 16 6 2 177 6 6 6 47 2 183 6 17 6 6 6 389 2 18 17 70


> %S A000001 1,2,2,4,2,6,2,8,4,6,2,16,2,6,6,16,2,16,2,16,
> %N A000001 Number of polynomials p with coefficients in {0,1} that divide x^n-1 and such that (x^n-1)/{p(x-1)} has all coefficients in {0,1}.
>  
> with differences at n = 12, 18, 20.
> 

  1 2 2 4 2 6 2 8 4 6 2 16 2 6 6 16 2 16 2 16 6 6 2 40 4 6 8 16 2 26 2 32 6 6 6 52 2 6 6 40 2 26 2 16 16 6 2 96 4 16 6 16 2 40 6 40 6 6 2 88 2 6 16 64 6 26 2 16 6 26 2 152 2 6 16 16 6 26 2 96 16 6 2 88 6 6 6 40 2 88 6 16 6 6 6 224 2 16 16 52

They differ for n = 12 18 20 24 28 30 36 40 42 44 45 48 50 52 54 56 60 63 66 68 70 72 75 76 78 80 84 88 90 92 96 98 99 100 102 104 105 108 110 112 114 116 117 120 124 126 130 132 135 136 138 140 144 147 148 150 152 153 154 156 160 162 164 165 168 170 171 172 174 175 176 180 182 184 186 188 189 190 192 195 196 198 200

which seems to be A102467 (without the first term).

Since A102467 is a complement of A102466, it seems that as soon as n belong to A102466 for every 01-polynomial p dividing x^n-1 its "dual" polynomial (x^n-1)/(p*(x-1)) is also a 01-polynomial.

Max