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A231713
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Square array A(i,j) = the sum of absolute values of digit differences in the matching positions of the factorial base representations of i and j, for i >= 0, j >= 0, read by antidiagonals.
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5
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0, 1, 1, 1, 0, 1, 2, 2, 2, 2, 2, 1, 0, 1, 2, 3, 3, 1, 1, 3, 3, 1, 2, 1, 0, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 0, 1, 2, 1, 2, 3, 3, 3, 3, 1, 1, 3, 3, 3, 3, 3, 2, 1, 2, 3, 0, 3, 2, 1, 2, 3, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 2, 3, 2, 1, 2, 3, 0, 3, 2, 1, 2, 3, 2, 3, 3, 3, 3, 3, 3, 1, 1, 3, 3, 3, 3, 3, 3, 3, 2, 3, 2, 1, 2, 1, 0, 1, 2, 1, 2, 3, 2, 3
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OFFSET
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0,7
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COMMENTS
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This table relates to the factorial base representation (A007623) in a similar way as A101080 relates to the binary system. See A230415 for another analog.
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LINKS
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FORMULA
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Each entry A(i,j) >= A230415(i,j) and also each entry A(i,j) >= abs(A230419(i,j)).
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EXAMPLE
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The top left corner of this square array begins as:
0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, ...
1, 0, 2, 1, 3, 2, 2, 1, 3, 2, 4, ...
1, 2, 0, 1, 1, 2, 2, 3, 1, 2, 2, ...
2, 1, 1, 0, 2, 1, 3, 2, 2, 1, 3, ...
2, 3, 1, 2, 0, 1, 3, 4, 2, 3, 1, ...
3, 2, 2, 1, 1, 0, 4, 3, 3, 2, 2, ...
1, 2, 2, 3, 3, 4, 0, 1, 1, 2, 2, ...
2, 1, 3, 2, 4, 3, 1, 0, 2, 1, 3, ...
2, 3, 1, 2, 2, 3, 1, 2, 0, 1, 1, ...
3, 2, 2, 1, 3, 2, 2, 1, 1, 0, 2, ...
3, 4, 2, 3, 1, 2, 2, 3, 1, 2, 0, ...
...
For example, A(1,2) = A(2,1) = 2 as 1 has factorial base representation '...0001' and 2 has factorial base representation '...0010', and adding the absolute values of the digit differences, we get 1+1 = 2.
On the other hand, A(3,5) = A(5,3) = 1, as 3 has factorial base representation '...0011' and 5 has factorial base representation '...0021', and they differ only by their second rightmost digit, the absolute value of difference being 1.
Note that as A007623(6)='100' and A007623(10)='120', we have A(6,10) = A(10,6) = 2.
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PROG
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(Scheme)
(define (A231713bi x y) (let loop ((x x) (y y) (i 2) (d 0)) (cond ((and (zero? x) (zero? y)) d) (else (loop (floor->exact (/ x i)) (floor->exact (/ y i)) (+ i 1) (+ d (abs (- (modulo x i) (modulo y i)))))))))
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CROSSREFS
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The topmost row and the leftmost column: A034968.
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KEYWORD
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AUTHOR
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STATUS
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approved
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