A276089 Factorial base representation of n is aerated (and then converted back to decimal): for n = sum_{i=1..} digit(i)*i! (with each digit(i) <= i), a(n) = sum_{i=1..} digit(i)*(2i-1)!.

0, 1, 6, 7, 12, 13, 120, 121, 126, 127, 132, 133, 240, 241, 246, 247, 252, 253, 360, 361, 366, 367, 372, 373, 5040, 5041, 5046, 5047, 5052, 5053, 5160, 5161, 5166, 5167, 5172, 5173, 5280, 5281, 5286, 5287, 5292, 5293, 5400, 5401, 5406, 5407, 5412, 5413, 10080, 10081, 10086, 10087, 10092, 10093, 10200, 10201, 10206, 10207, 10212, 10213, 10320, 10321

Offset: 0

Author

Antti Karttunen, Aug 19 2016

Keywords

• nonn
• base

Comments

Here "aeration" means inserting zeros between the digits of factorial base representation of n. See the examples.

Examples

			   n A007623(n)  "aerated"   and converted back to decimal gives
                             a(n)
   0       0         0         0
   1       1         1         1
   2      10       100         6
   3      11       101         7
   4      20       200        12
   5      21       201        13
   6     100     10000       120
   7     101     10001       121
   8     110     10100       126
   9     111     10101       127
  10     120     10200       132
  11     121     10201       133
  12     200     20000       240
  13     201     20001       241
  14     210     20100       246
  15     211     20101       247
  16     220     20200       252
  17     221     20201       253
  18     300     30000       360
  19     301     30001       361
  20     310     30100       366
  21     311     30101       367
  22     320     30200       372
  23     321     30201       373
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..5039
• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A000142, A007623, A084558

Cf. A276090 (a left inverse).

Cf. A275959 (subsequence).

Programs

Scheme

;; Standalone program:
(define (A276089 n) (let loop ((n n) (s 0) (f 1) (i 2) (j 2)) (if (zero? n) s (let ((d (modulo n i))) (loop (/ (- n d) i) (+ s (* f d)) (* j (+ 1 j) f) (+ 1 i) (+ 2 j))))))

Formula

When n = sum_{i=1..A084558(n)} d(i)*i! (which is unique representation as long as each d(i) <= i) then a(n) = sum_{i=1..A084558(n)} d(i)*(2i-1)!, where A084558(n) gives the length of factorial base representation of n.

Other identities. For all n >= 0:

A276090(a(n)) = n.