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A308496
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Numbers with digits 1,2,4,7 when written in base 8.
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1
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1, 2, 4, 7, 9, 10, 12, 15, 17, 18, 20, 23, 33, 34, 36, 39, 57, 58, 60, 63, 73, 74, 76, 79, 81, 82, 84, 87, 97, 98, 100, 103, 121, 122, 124, 127, 137, 138, 140, 143, 145, 146, 148, 151, 161, 162, 164, 167, 185, 186, 188, 191, 265, 266, 268, 271, 273
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OFFSET
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1,2
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COMMENTS
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For any natural number n, the set of terms of this sequence between indices (4^n-1)/3 and (4^(n+1)-4)/3 is "isomorphic" to the group of n-th order floretions. In this case, group multiplication is given by bitwise logical operations (see EXAMPLE). Note that the case of n = 1 is simply the quaternions.
In the table below, the left column is the binary representation, the middle column the terms of ((a(n)) and the right column the conventional notation. Multiply x*y (disregarding signs) using the bitwise XNOR operation, where x and y are any floretions of the same order. The XNOR operation returns a 1 if the number of 1's in its inputs is even, and a 0 if the number of 1's is odd. This operation is used to calculate the base vector of the result.
**** 1st-order floretions (= quaternions) ****
| binary | decimal | floretion
1 | 001 | 1 | i
2 | 010 | 2 | j
4 | 100 | 4 | k
7 | 111 | 7 | e (unit)
**** 2nd-order floretions ****
1_1 | 001_001 | 9 | ii
1_2 | 001_010 | 10 | ij
1_4 | 001_100 | 12 | ik
1_7 | 001_111 | 15 | ie
2_1 | 010_001 | 17 | ji
2_2 | 010_010 | 18 | jj
2_4 | 010_100 | 20 | jk
2_7 | 010_111 | 23 | je
4_1 | 100_001 | 33 | ki
4_2 | 100_010 | 34 | kj
4_4 | 100_100 | 36 | kk
4_7 | 100_111 | 39 | ke
7_1 | 111_001 | 57 | ei
7_2 | 111_010 | 58 | ej
7_4 | 111_100 | 60 | ek
7_7 | 111_111 | 63 | ee
**** 3rd-order floretions ****
1_1_1
1_1_2
...
Note that for a floretion of order n, two digits from any one of its "binary triplets" _abc_ determine the other since XOR(a,b,c) = 1.
When working with a floretion algebra over the reals, i.e., elements of the form x = q_1*f_1 + ... q_m*f_m where q_1,...,q_m are real numbers and f_1,...,f_m are any floretions of the same order, then x may also be referred to as a "floretion". In this case f_1,...,f_m (i.e., terms of this sequence) may be referred to as "floretion base vectors" to avoid confusion.
Taking signs into account:
Given two binary representations (ab) and (cd) for quaternion elements, define multiplication as:
Compute (XNOR(a,c))(XNOR(b,d)) to get the base vector of the result.
Compute AND(b,c), AND(XNOR(a,b),d), and AND(a,XNOR(c,d)). These are all bitwise AND operations.
The sign is negative if and only if the total number of 1's in the results is even.
For example, with k*j = (10)*(01) = -i, compute:
The base vector as (XNOR(1,0) XNOR(0,1)) = (0)(0) = i.
The signs as AND(0,1), AND(XNOR(1,0),1), AND(1, XNOR(0,1)) = 0, 0, 0. There are zero 1's in total, which is an even number, so the result is negative.
An example of image processing: take for example a quaternion x = .2i + .5j + .3k + e. Assume we have a square monitor (aspect ratio). Furthermore, assume the screen is divided into 4 squares- one for i (bottom left), one for j (top left), one for k (top right) one for e (bottom right) and that the coefficient is the amount the pixels are lit up on the screen (1 being full brightness, 0 being off- this could be modified later to accomodate negative numbers). Now imagine we have square monitor of resolution 2^n x 2^n. Then we can represent any black and white image with that resolution with an n-th order floretion. This means we can multiply images together, with some parallels to Fourier analysis.
Multiplying an image by an idempotent floretion would allow one to repeatedly apply a specific transformation (e.g., a rotation, scaling, or some other operation) to an image, and then undo all of those transformations by continuing to apply the same operation a certain number of times. It could be used in applications such as data encryption, where an image could be "scrambled" using a specific floretion and then "unscrambled" by continuing to apply the same floretion.
A compact definition of multiplication is x*y = (ab)(cd) = (-1)^{m+1} (aqc)(bqd) where m = b&c + (aqb)&d + a&(cqd) and "q", "&" are the bitwise XNOR and AND operators respectively. - Creighton Dement, Jul 09 2023
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LINKS
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MATHEMATICA
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A308496Q[n_]:=ContainsOnly[IntegerDigits[n, 8], {1, 2, 4, 7}];
Select[Range[1000], A308496Q] (* Paolo Xausa, Dec 31 2023 *)
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PROG
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(PARI) is(n)=!#setminus(Set(digits(n, 8)), [1, 2, 4, 7]);
a(n) =
{
local(total_count, index);
until(total_count == n+1, if(is(index)==1, total_count++); index++);
index-1;
}
(PARI) a(n, b=8, d=[1, 2, 4, 7]) = { for (w=1, oo, if (n>#d^w, n-=#d^w, return (fromdigits(apply(x -> d[1+x], digits(#d^w+n-1, #d))[2..-1], b)))) } \\ Rémy Sigrist, Jun 01 2019
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CROSSREFS
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KEYWORD
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nonn,base,uned
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AUTHOR
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STATUS
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approved
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