A230415 - OEIS

%I #21 Sep 09 2017 19:33:47

%S 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,

%T 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,

%U 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

%N 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.

%C 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.

%H Antti Karttunen, <a href="/A230415/b230415.txt">The first 121 antidiagonals of the table, flattened</a>

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

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

%e The top left corner of this square array begins as:

%e 0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, ...

%e 1, 0, 2, 1, 2, 1, 2, 1, 3, 2, 3, ...

%e 1, 2, 0, 1, 1, 2, 2, 3, 1, 2, 2, ...

%e 2, 1, 1, 0, 2, 1, 3, 2, 2, 1, 3, ...

%e 1, 2, 1, 2, 0, 1, 2, 3, 2, 3, 1, ...

%e 2, 1, 2, 1, 1, 0, 3, 2, 3, 2, 2, ...

%e 1, 2, 2, 3, 2, 3, 0, 1, 1, 2, 1, ...

%e 2, 1, 3, 2, 3, 2, 1, 0, 2, 1, 2, ...

%e 2, 3, 1, 2, 2, 3, 1, 2, 0, 1, 1, ...

%e 3, 2, 2, 1, 3, 2, 2, 1, 1, 0, 2, ...

%e 2, 3, 2, 3, 1, 2, 1, 2, 1, 2, 0, ...

%e ...

%e 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.

%e 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.

%e 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.

%t 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 *)

%o (Scheme)

%o (define (A230415 n) (A230415bi (A025581 n) (A002262 n)))

%o (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)))))))

%Y The topmost row and the leftmost column: A060130.

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

%K nonn,base,tabl

%O 0,7

%A _Antti Karttunen_, Nov 10 2013