login
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.
5
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.
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
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, Dec 18 2016
STATUS
approved