This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A279732 #16 Jan 06 2017 13:01:28
%S A279732 1,2,6,8,24,30,48,120,240,720,840,1440,1560,5040,10080,15120,40320,
%T A279732 45360,80640,120960,362880,403200,725760,1088640,3628800,3991680,
%U A279732 7257600,7620480,10886400,39916800,43545600,79833600,119750400,159667200,479001600,958003200
%N 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.
%C A279732 This sequence is to factorial base what A278742 is to base 10.
%C A279732 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, ...
%C A279732 Occasionally, the sum of the first n terms equals A033312(k) for some k;
%C A279732 - In that case: a(n+1)=k!, and k! divides a(m) for any m>n,
%C A279732 - 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, ...
%C A279732 - Conjecture: this happens infinitely often.
%H A279732 Rémy Sigrist, <a href="/A279732/b279732.txt">Table of n, a(n) for n = 1..10000</a>
%H A279732 Rémy Sigrist, <a href="/A279732/a279732.gp.txt">PARI program for A279732</a>
%H A279732 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%e A279732 The first terms in base 10 and factorial base, alongside their partial sums in factorial base, are:
%e A279732 n a(n) a(n) in fact. base Partial sum in fact. base
%e A279732 -- --------- --------------------- -------------------------
%e A279732 1 1 1 1
%e A279732 2 2 1,0 1,1
%e A279732 3 6 1,0,0 1,1,1
%e A279732 4 8 1,1,0 2,2,1
%e A279732 5 24 1,0,0,0 1,2,2,1
%e A279732 6 30 1,1,0,0 2,3,2,1
%e A279732 7 48 2,0,0,0 4,3,2,1
%e A279732 8 120 1,0,0,0,0 1,4,3,2,1
%e A279732 9 240 2,0,0,0,0 3,4,3,2,1
%e A279732 10 720 1,0,0,0,0,0 1,3,4,3,2,1
%e A279732 11 840 1,1,0,0,0,0 2,4,4,3,2,1
%e A279732 12 1440 2,0,0,0,0,0 4,4,4,3,2,1
%e A279732 13 1560 2,1,0,0,0,0 6,5,4,3,2,1
%e A279732 14 5040 1,0,0,0,0,0,0 1,6,5,4,3,2,1
%e A279732 15 10080 2,0,0,0,0,0,0 3,6,5,4,3,2,1
%e A279732 16 15120 3,0,0,0,0,0,0 6,6,5,4,3,2,1
%e A279732 17 40320 1,0,0,0,0,0,0,0 1,6,6,5,4,3,2,1
%e A279732 18 45360 1,1,0,0,0,0,0,0 2,7,6,5,4,3,2,1
%e A279732 19 80640 2,0,0,0,0,0,0,0 4,7,6,5,4,3,2,1
%e A279732 20 120960 3,0,0,0,0,0,0,0 7,7,6,5,4,3,2,1
%e A279732 21 362880 1,0,0,0,0,0,0,0,0 1,7,7,6,5,4,3,2,1
%e A279732 22 403200 1,1,0,0,0,0,0,0,0 2,8,7,6,5,4,3,2,1
%e A279732 23 725760 2,0,0,0,0,0,0,0,0 4,8,7,6,5,4,3,2,1
%e A279732 24 1088640 3,0,0,0,0,0,0,0,0 7,8,7,6,5,4,3,2,1
%e A279732 25 3628800 1,0,0,0,0,0,0,0,0,0 1,7,8,7,6,5,4,3,2,1
%e A279732 26 3991680 1,1,0,0,0,0,0,0,0,0 2,8,8,7,6,5,4,3,2,1
%e A279732 27 7257600 2,0,0,0,0,0,0,0,0,0 4,8,8,7,6,5,4,3,2,1
%e A279732 28 7620480 2,1,0,0,0,0,0,0,0,0 6,9,8,7,6,5,4,3,2,1
%e A279732 29 10886400 3,0,0,0,0,0,0,0,0,0 9,9,8,7,6,5,4,3,2,1
%e A279732 30 39916800 1,0,0,0,0,0,0,0,0,0,0 1,9,9,8,7,6,5,4,3,2,1
%e A279732 31 43545600 1,1,0,0,0,0,0,0,0,0,0 2,10,9,8,7,6,5,4,3,2,1
%e A279732 32 79833600 2,0,0,0,0,0,0,0,0,0,0 4,10,9,8,7,6,5,4,3,2,1
%e A279732 33 119750400 3,0,0,0,0,0,0,0,0,0,0 7,10,9,8,7,6,5,4,3,2,1
%e A279732 34 159667200 4,0,0,0,0,0,0,0,0,0,0 11,10,9,8,7,6,5,4,3,2,1
%t A279732 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 *)
%Y A279732 Cf. A000142, A007623, A033312, A278742, A278743.
%K A279732 nonn,base
%O A279732 1,2
%A A279732 _Rémy Sigrist_, Dec 18 2016