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

Author

Rémy Sigrist, Dec 18 2016

Keywords

• nonn
• base

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.

Examples

			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
		

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

Crossrefs

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

Programs

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 *)