A279732 Lexicographically least strictly increasing sequence such that, for any n>0, Sum_{k=1..n} a(k) can be computed without carries in factorial base.

1, 2, 6, 8, 24, 30, 48, 120, 240, 720, 840, 1440, 1560, 5040, 10080, 15120, 40320, 45360, 80640, 120960, 362880, 403200, 725760, 1088640, 3628800, 3991680, 7257600, 7620480, 10886400, 39916800, 43545600, 79833600, 119750400, 159667200, 479001600, 958003200

Offset

1,2

Comments

This sequence is to factorial base what A278742 is to base 10.

This sequence contains the factorial numbers (A000142); the corresponding indices are 1, 2, 3, 5, 8, 10, 14, 17, 21, 25, 30, 35, 39, 45, 49, 56, 62, 67, 74, 79, 87, 93, 102, 108, 116, 122, 131, 138, 148, 155, ...

Occasionally, the sum of the first n terms equals A033312(k) for some k;

- In that case: a(n+1)=k!, and k! divides a(m) for any m>n,

- The corresponding indices are 1, 7, 13, 34, 44, 61, 73, 101, 115, 147, 343, 387, 487, 605, 657, 788, 1226, 1296, 1575, 2986, 3586, 5152, 5260, 8236, 9173, ...

- Conjecture: this happens infinitely often.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, PARI program for A279732

Index entries for sequences related to factorial base representation

Example

The first terms in base 10 and factorial base, alongside their partial sums in factorial base, are:
n    a(n)        a(n) in fact. base      Partial sum in fact. base
--   ---------   ---------------------   -------------------------
1            1                       1                         1
2            2                     1,0                       1,1
3            6                   1,0,0                     1,1,1
4            8                   1,1,0                     2,2,1
5           24                 1,0,0,0                   1,2,2,1
6           30                 1,1,0,0                   2,3,2,1
7           48                 2,0,0,0                   4,3,2,1
8          120               1,0,0,0,0                 1,4,3,2,1
9          240               2,0,0,0,0                 3,4,3,2,1
10         720             1,0,0,0,0,0               1,3,4,3,2,1
11         840             1,1,0,0,0,0               2,4,4,3,2,1
12        1440             2,0,0,0,0,0               4,4,4,3,2,1
13        1560             2,1,0,0,0,0               6,5,4,3,2,1
14        5040           1,0,0,0,0,0,0             1,6,5,4,3,2,1
15       10080           2,0,0,0,0,0,0             3,6,5,4,3,2,1
16       15120           3,0,0,0,0,0,0             6,6,5,4,3,2,1
17       40320         1,0,0,0,0,0,0,0           1,6,6,5,4,3,2,1
18       45360         1,1,0,0,0,0,0,0           2,7,6,5,4,3,2,1
19       80640         2,0,0,0,0,0,0,0           4,7,6,5,4,3,2,1
20      120960         3,0,0,0,0,0,0,0           7,7,6,5,4,3,2,1
21      362880       1,0,0,0,0,0,0,0,0         1,7,7,6,5,4,3,2,1
22      403200       1,1,0,0,0,0,0,0,0         2,8,7,6,5,4,3,2,1
23      725760       2,0,0,0,0,0,0,0,0         4,8,7,6,5,4,3,2,1
24     1088640       3,0,0,0,0,0,0,0,0         7,8,7,6,5,4,3,2,1
25     3628800     1,0,0,0,0,0,0,0,0,0       1,7,8,7,6,5,4,3,2,1
26     3991680     1,1,0,0,0,0,0,0,0,0       2,8,8,7,6,5,4,3,2,1
27     7257600     2,0,0,0,0,0,0,0,0,0       4,8,8,7,6,5,4,3,2,1
28     7620480     2,1,0,0,0,0,0,0,0,0       6,9,8,7,6,5,4,3,2,1
29    10886400     3,0,0,0,0,0,0,0,0,0       9,9,8,7,6,5,4,3,2,1
30    39916800   1,0,0,0,0,0,0,0,0,0,0     1,9,9,8,7,6,5,4,3,2,1
31    43545600   1,1,0,0,0,0,0,0,0,0,0    2,10,9,8,7,6,5,4,3,2,1
32    79833600   2,0,0,0,0,0,0,0,0,0,0    4,10,9,8,7,6,5,4,3,2,1
33   119750400   3,0,0,0,0,0,0,0,0,0,0    7,10,9,8,7,6,5,4,3,2,1
34   159667200   4,0,0,0,0,0,0,0,0,0,0   11,10,9,8,7,6,5,4,3,2,1

Mathematica

r = MixedRadix[Reverse@ Range[2, 30]]; f[a_] := Function[w, Function[s, Total@ Map[PadLeft[#, s] &, w]]@ Max@ Map[Length, w]]@ Map[IntegerDigits[#, r] &, a]; g[w_] := Times @@ Boole@ MapIndexed[#1 <= First@ #2 &, Reverse@ w] > 0; a = {1}; Do[k = Max@ a + 1; While[! g@ f@ Join[a, {k}], k++]; AppendTo[a, k], {n, 2, 16}]; a (* Michael De Vlieger, Dec 18 2016 *)

Crossrefs

Cf. A000142, A007623, A033312, A278742, A278743.

Sequence in context: A334898 A081957 A334901 * A260669 A122758 A122756

Adjacent sequences: A279729 A279730 A279731 * A279733 A279734 A279735

Keyword

nonn,base

Author

Rémy Sigrist, Dec 18 2016

Status

approved