A275730 Square array A(n,d): overwrite with zero the digit at position d from right (indicating radix d+2) in the factorial base representation of n, then convert back to decimal, read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), etc.

0, 0, 0, 0, 1, 2, 0, 1, 0, 2, 0, 1, 2, 1, 4, 0, 1, 2, 3, 0, 4, 0, 1, 2, 3, 4, 1, 6, 0, 1, 2, 3, 4, 5, 6, 6, 0, 1, 2, 3, 4, 5, 0, 7, 8, 0, 1, 2, 3, 4, 5, 6, 1, 6, 8, 0, 1, 2, 3, 4, 5, 6, 7, 2, 7, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 3, 6, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 4, 7, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 12, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 13, 14

Offset: 0

Author

Antti Karttunen, Aug 08 2016

Keywords

• nonn
• base
• tabl

Examples

			Columns 0-4 of rows 0 - 24 of the array:
  0, 0, 0, 0, 0, ... [No matter which digit of zero we clear, it stays zero forever]
  0, 1, 1, 1, 1  ... [When clearing the least significant digit (pos. 0) of one, "1", we get zero, and clearing any other digit past the most significant digit keeps one as one]
2, 0, 2, 2, 2, ... [Clearing the least significant digit of 2, "10", doesn't affect it, but clearing the digit-1 zeros the whole number].
  2, 1, 3, 3, 3, ... [Clearing the least significant factorial base digit of 3 ("11") gives "10", 2, clearing the digit-1 gives "01" = 1, and clearing any digit past the most significant keeps "11" as it is, 3].
  4, 0, 4, 4, 4
  4, 1, 5, 5, 5
  6, 6, 0, 6, 6
  6, 7, 1, 7, 7
  8, 6, 2, 8, 8
  8, 7, 3, 9, 9
  10, 6, 4, 10, 10
  10, 7, 5, 11, 11
  12, 12, 0, 12, 12
  12, 13, 1, 13, 13
  14, 12, 2, 14, 14
  14, 13, 3, 15, 15
  16, 12, 4, 16, 16
  16, 13, 5, 17, 17
  18, 18, 0, 18, 18
  18, 19, 1, 19, 19
  20, 18, 2, 20, 20
  20, 19, 3, 21, 21
  22, 18, 4, 22, 22
  22, 19, 5, 23, 23
  24, 24, 24, 0, 24
  ...
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..7259; the first 120 antidiagonals of array
• Index entries for sequences related to factorial base representation

Crossrefs

Transpose: A275731.

Column 0: A052928, Main diagonal: A001477.

Cf. A084558, A257687.

Can be used when computing A275732 and A275736.

Programs

Scheme

(define (A275730 n) (A275730bi (A002262 n) (A025581 n)))
(define (A275730bi n c) (let loop ((z 0) (n n) (m 2) (f 1) (c c)) (let ((d (modulo n m))) (cond ((zero? n) z) ((zero? c) (loop z (/ (- n d) m) (+ 1 m) (* f m) (- c 1))) (else (loop (+ z (* f d)) (/ (- n d) m) (+ 1 m) (* f m) (- c 1)))))))

Formula

Other identities:

For all n >= 1, A(n,A084558(n)-1) = A257687(n).

For all n >= 0, A(n,A084558(n)) = n.