A230415 Square array T(i,j) giving the number of differing digits in the factorial base representations of i and j, for i >= 0, j >= 0, read by antidiagonals.

0, 1, 1, 1, 0, 1, 2, 2, 2, 2, 1, 1, 0, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 0, 1, 1, 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, 2, 2, 1, 2, 2, 0, 2, 2, 1, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 1, 2, 2, 1, 2, 2, 0, 2, 2, 1, 2, 2, 1, 2, 2, 3, 3, 3, 3, 1, 1, 3, 3, 3, 3, 2, 2, 2, 1, 2, 2, 1, 2, 1, 0, 1, 2, 1, 2, 2, 1, 2

Offset

0,7

Comments

This table relates to the factorial base representation (A007623) in a somewhat similar way as A101080 relates to the binary system. See A231713 for another analog.

Links

Antti Karttunen, The first 121 antidiagonals of the table, flattened

Formula

T(n,0) = T(0,n) = A060130(n).

Each entry T(i,j) <= A231713(i,j).

Example

The top left corner of this square array begins as:
0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, ...
1, 0, 2, 1, 2, 1, 2, 1, 3, 2, 3, ...
1, 2, 0, 1, 1, 2, 2, 3, 1, 2, 2, ...
2, 1, 1, 0, 2, 1, 3, 2, 2, 1, 3, ...
1, 2, 1, 2, 0, 1, 2, 3, 2, 3, 1, ...
2, 1, 2, 1, 1, 0, 3, 2, 3, 2, 2, ...
1, 2, 2, 3, 2, 3, 0, 1, 1, 2, 1, ...
2, 1, 3, 2, 3, 2, 1, 0, 2, 1, 2, ...
2, 3, 1, 2, 2, 3, 1, 2, 0, 1, 1, ...
3, 2, 2, 1, 3, 2, 2, 1, 1, 0, 2, ...
2, 3, 2, 3, 1, 2, 1, 2, 1, 2, 0, ...
...
For example, T(1,2) = T(2,1) = 2 as 1 has factorial base representation '...0001' and 2 has factorial base representation '...0010', and they differ by their two least significant digits.
On the other hand, T(3,5) = T(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.
Note that as A007623(6)='100' and A007623(10)='120', we have T(6,10) = T(10,6) = 1 (instead of 2 as in A231713, cf. also its Example section), as here we count only the number of differing digit positions, but ignore the magnitudes of their differences.

Mathematica

nn = 14; m = 1; While[m! < nn, m++]; m; Table[Function[w, Count[Subtract @@ Map[PadLeft[#, Max@ Map[Length, w]] &, w], k_ /; k != 0]]@ Map[IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]] &, {i - j, j}], {i, 0, nn}, {j, 0, i}] // Flatten (* Michael De Vlieger, Jun 27 2016, Version 10.2 *)

Prog

(Scheme)
(define (A230415 n) (A230415bi (A025581 n) (A002262 n)))
(define (A230415bi 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 (if (= (modulo x i) (modulo y i)) 0 1)))))))

Crossrefs

The topmost row and the leftmost column: A060130.

Only the lower triangular region: A230417. Related arrays: A230419, A231713. Cf. also A101080, A084558, A230410.

Sequence in context: A049241 A321858 A334235 * A101080 A130836 A279185

Adjacent sequences: A230412 A230413 A230414 * A230416 A230417 A230418

Keyword

nonn,base,tabl

Author

Antti Karttunen, Nov 10 2013

Status

approved