|
| |
|
|
A219256
|
|
Numbers whose base 64 representation does not contain any of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 16, 17, 18, 24, 25, 32, 34, 36, 40, 44, 48, 50.
|
|
0
|
|
|
|
10, 11, 13, 14, 15, 19, 20, 21, 22, 23, 26, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 650, 651, 653, 654, 655, 659, 660, 661, 662, 663, 666, 667, 668, 669, 670, 671, 673, 675, 677, 678, 679, 681, 682, 683, 685, 686, 687, 689, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
Numbers whose base 64 representation consists only of digits {10, 11, 13, 14, 15, 19, 20, 21, 22, 23, 26, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63}.
Positive integers all of whose digits are 3-dimensional when using the following base 64 representation by subsets of the edges of a (regular) tetrahedron.
Define an alternate (regarding appearance of digits only) base 64 representation system as follows: Let ABCD be a (regular) tetrahedron oriented such that the base triangle ABC faces the viewer, and the edge AC, with A on the left, is horizontal. Define the base 64 digits corresponding to decimal 1, 2, 4, 8, 16, and 32 as the oriented edges AC, AB, BC, CD, AD, BD, respectively. (This scheme (of 6! = 720 similar schemes) assigns these values clockwise beginning with the base triangle's edges then continuing with the edges containing the fourth vertex.) The remaining positive base 64 digits are each defined as the unique geometric figure that is the union of two or more of the six oriented edges having the corresponding sum. For example, triangle ABC, the union of edges AC, AB, and BC, is the digit for 1 + 2 + 4 = 7. The entire tetrahedron is thus the digit for 1 + 2 + 4 + 8 + 16 + 32 = 63. The digit for 0 is a blank space by this method. (Variants with visible zeros are certainly possible.) Of the nonzero digits, 41 are 3-dimensional, 16 are 2-dimensional, and 6 are 1-dimensional. (Widths and depths of physically depicted line segments are ignored.) Of course all these digits could also be represented in two or fewer dimensions by drawing their projections onto, say, the plane of the fixed face ABC (thus depicting a top view), in which case each of these 64 digits is still distinct.
Base 64 representations share ease of conversion to and from binary with octal and hexadecimal (and any base 2^n for n > 1) notations. If a similar system were employed with the cube or octahedron, the base would be 4096 (=2^12); with the dodecahedron or icosahedron the base would be 1073741824 (=2^30). Other bases of form 2^(2n) are certainly possible with n-gonal pyramids. For bases 2^n, n-gons could be used. More arbitrary shapes could also be used in this manner. Finally, similar systems could be devised using vertices only (instead of just edges) or both vertices and edges in defining the appearance of digits (glyphs).
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
The number 10 is a term since its single-digit representation consists of segments of two skew lines, thus 3-dimensional. The number 652 = 10*64 + 12 is not a term: Although the digit for 10 is 3-dimensional, the digit for 12 is an angle, so only 2-dimensional.
The number 14836941 = 56*64^3 + 38*64^2 + 19*64 + 13 is a term since each of the four digits 56, 38, 19, and 13 is a 3-dimensional figure resulting from removing the edges belonging to one of the tetrahedron's four faces. Note also that (56)(38)(19)(13) base 64 <==> 111000 base 2, 100110 base 2, 010011 base 2, 001101 base 2 <==> 111000100110010011001101 base 2 <==> 14836941 base 10.
|
|
|
MATHEMATICA
|
Flatten[Table[FromDigits[#, 64]&/@Tuples[{10, 11, 13, 14, 15, 19, 20, 21, 22, 23, 26, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63}, n], {n, 2}]]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,base
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
