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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)!.
4
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
0,3
COMMENTS
Here "aeration" means inserting zeros between the digits of factorial base representation of n. See the examples.
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.
EXAMPLE
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
PROG
(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))))))
CROSSREFS
Cf. A276090 (a left inverse).
Cf. A275959 (subsequence).
Sequence in context: A374920 A191337 A352090 * A037364 A315833 A155942
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Aug 19 2016
STATUS
approved