|
| |
|
|
A329327
|
|
Numbers whose binary expansion has Lyndon factorization of length 2 (the minimum for n > 1).
|
|
6
|
|
|
|
2, 3, 5, 9, 11, 17, 19, 23, 33, 35, 37, 39, 43, 47, 65, 67, 69, 71, 75, 77, 79, 87, 95, 129, 131, 133, 135, 137, 139, 141, 143, 147, 149, 151, 155, 157, 159, 171, 175, 183, 191, 257, 259, 261, 263, 265, 267, 269, 271, 275, 277, 279, 281, 283, 285, 287, 293
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
First differs from A329357 in having 77 and lacking 83.
Also numbers whose decapitated binary expansion is a Lyndon word (see also A329401).
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
EXAMPLE
|
The binary expansion of each term together with its Lyndon factorization begins:
2: (10) = (1)(0)
3: (11) = (1)(1)
5: (101) = (1)(01)
9: (1001) = (1)(001)
11: (1011) = (1)(011)
17: (10001) = (1)(0001)
19: (10011) = (1)(0011)
23: (10111) = (1)(0111)
33: (100001) = (1)(00001)
35: (100011) = (1)(00011)
37: (100101) = (1)(00101)
39: (100111) = (1)(00111)
43: (101011) = (1)(01011)
47: (101111) = (1)(01111)
65: (1000001) = (1)(000001)
67: (1000011) = (1)(000011)
69: (1000101) = (1)(000101)
71: (1000111) = (1)(000111)
75: (1001011) = (1)(001011)
77: (1001101) = (1)(001101)
|
|
|
MATHEMATICA
|
lynQ[q_]:=Array[Union[{q, RotateRight[q, #]}]=={q, RotateRight[q, #]}&, Length[q]-1, 1, And];
lynfac[q_]:=If[Length[q]==0, {}, Function[i, Prepend[lynfac[Drop[q, i]], Take[q, i]]][Last[Select[Range[Length[q]], lynQ[Take[q, #1]]&]]]];
Select[Range[100], Length[lynfac[IntegerDigits[#, 2]]]==2&]
|
|
|
CROSSREFS
|
Positions of rows of length 2 in A329314.
The "co-" and reversed version is A329357.
Length of the co-Lyndon factorization of the binary expansion is A329312.
Cf. A059966, A060223, A211097, A275692, A328594, A328595, A328596, A329131, A329313, A329325, A329326, A339608.
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
