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%I #15 Jan 05 2024 12:29:34
%S 0,1,1,2,0,2,3,3,3,3,4,2,4,2,4,5,5,5,5,5,5,6,4,0,4,0,4,6,7,7,1,1,1,1,
%T 7,7,8,6,8,0,2,0,8,6,8,9,9,9,9,3,3,9,9,9,9,10,8,10,8,10,2,10,8,10,8,
%U 10,11,11,11,11,11,11,11,11,11,11,11,11
%N Square array A(n, k), n, k >= 0, read by antidiagonals; the factorial base expansion of A(n, k) is obtained by adding componentwise and reducing modulo their radix the digits of the factorial base expansions of n and k.
%C The nonnegative integers together with A form an abelian group; A225901 gives inverse elements.
%C Each row is a permutation of the nonnegative integers.
%H Andrew Howroyd, <a href="/A354438/b354438.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals)
%H Rémy Sigrist, <a href="/A354438/a354438.png">Colored representation of the array A(n, k) for n, k < 7!</a> (the hue is function of A(n, k), black pixels correspond to 0's)
%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%F A(n, k) = A(k, n).
%F A(m, A(n, k)) = A(A(m, n), k).
%F A(n, 0) = n.
%F A(n, k) = 0 iff k = A225901(n).
%F A(n, 1) = A004442(n).
%e Square array A(n, k) begins:
%e n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
%e ---+----------------------------------------------------------------
%e 0| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
%e 1| 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14
%e 2| 2 3 4 5 0 1 8 9 10 11 6 7 14 15 16 17
%e 3| 3 2 5 4 1 0 9 8 11 10 7 6 15 14 17 16
%e 4| 4 5 0 1 2 3 10 11 6 7 8 9 16 17 12 13
%e 5| 5 4 1 0 3 2 11 10 7 6 9 8 17 16 13 12
%e 6| 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
%e 7| 7 6 9 8 11 10 13 12 15 14 17 16 19 18 21 20
%e 8| 8 9 10 11 6 7 14 15 16 17 12 13 20 21 22 23
%e 9| 9 8 11 10 7 6 15 14 17 16 13 12 21 20 23 22
%e 10| 10 11 6 7 8 9 16 17 12 13 14 15 22 23 18 19
%e 11| 11 10 7 6 9 8 17 16 13 12 15 14 23 22 19 18
%e 12| 12 13 14 15 16 17 18 19 20 21 22 23 0 1 2 3
%e 13| 13 12 15 14 17 16 19 18 21 20 23 22 1 0 3 2
%e 14| 14 15 16 17 12 13 20 21 22 23 18 19 2 3 4 5
%e 15| 15 14 17 16 13 12 21 20 23 22 19 18 3 2 5 4
%o (PARI) A(n,k, s=i->i+1) = { my (v=0, f=1, r); for (i=1, oo, if (n==0 && k==0, return (v), r=s(i); v+=f*((n+k)%r); f*=r; n\=r; k\=r)) }
%Y Cf. A003987, A004442, A108731, A225901, A354470 (primorial base analog).
%K nonn,tabl,base
%O 0,4
%A _Rémy Sigrist_, May 28 2022