A276089 - OEIS

%I #12 Aug 21 2016 17:45:02

%S 0,1,6,7,12,13,120,121,126,127,132,133,240,241,246,247,252,253,360,

%T 361,366,367,372,373,5040,5041,5046,5047,5052,5053,5160,5161,5166,

%U 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

%N 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)!.

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

%H Antti Karttunen, <a href="/A276089/b276089.txt">Table of n, a(n) for n = 0..5039</a>

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>

%F 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.

%F Other identities. For all n >= 0:

%F A276090(a(n)) = n.

%e    n A007623(n)  "aerated"   and converted back to decimal gives

%e                              a(n)

%e    0       0         0         0

%e    1       1         1         1

%e    2      10       100         6

%e    3      11       101         7

%e    4      20       200        12

%e    5      21       201        13

%e    6     100     10000       120

%e    7     101     10001       121

%e    8     110     10100       126

%e    9     111     10101       127

%e   10     120     10200       132

%e   11     121     10201       133

%e   12     200     20000       240

%e   13     201     20001       241

%e   14     210     20100       246

%e   15     211     20101       247

%e   16     220     20200       252

%e   17     221     20201       253

%e   18     300     30000       360

%e   19     301     30001       361

%e   20     310     30100       366

%e   21     311     30101       367

%e   22     320     30200       372

%e   23     321     30201       373

%o (Scheme)

%o ;; Standalone program:

%o (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))))))

%Y Cf. A000142, A007623, A084558

%Y Cf. A276090 (a left inverse).

%Y Cf. A275959 (subsequence).

%K nonn,base

%O 0,3

%A _Antti Karttunen_, Aug 19 2016