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%I #16 Jan 06 2017 13:01:28
%S 1,2,6,8,24,30,48,120,240,720,840,1440,1560,5040,10080,15120,40320,
%T 45360,80640,120960,362880,403200,725760,1088640,3628800,3991680,
%U 7257600,7620480,10886400,39916800,43545600,79833600,119750400,159667200,479001600,958003200
%N 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 This sequence is to factorial base what A278742 is to base 10.
%C 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 Occasionally, the sum of the first n terms equals A033312(k) for some k;
%C - In that case: a(n+1)=k!, and k! divides a(m) for any m>n,
%C - 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 - Conjecture: this happens infinitely often.
%H Rémy Sigrist, <a href="/A279732/b279732.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A279732/a279732.gp.txt">PARI program for A279732</a>
%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%e The first terms in base 10 and factorial base, alongside their partial sums in factorial base, are:
%e n a(n) a(n) in fact. base Partial sum in fact. base
%e -- --------- --------------------- -------------------------
%e 1 1 1 1
%e 2 2 1,0 1,1
%e 3 6 1,0,0 1,1,1
%e 4 8 1,1,0 2,2,1
%e 5 24 1,0,0,0 1,2,2,1
%e 6 30 1,1,0,0 2,3,2,1
%e 7 48 2,0,0,0 4,3,2,1
%e 8 120 1,0,0,0,0 1,4,3,2,1
%e 9 240 2,0,0,0,0 3,4,3,2,1
%e 10 720 1,0,0,0,0,0 1,3,4,3,2,1
%e 11 840 1,1,0,0,0,0 2,4,4,3,2,1
%e 12 1440 2,0,0,0,0,0 4,4,4,3,2,1
%e 13 1560 2,1,0,0,0,0 6,5,4,3,2,1
%e 14 5040 1,0,0,0,0,0,0 1,6,5,4,3,2,1
%e 15 10080 2,0,0,0,0,0,0 3,6,5,4,3,2,1
%e 16 15120 3,0,0,0,0,0,0 6,6,5,4,3,2,1
%e 17 40320 1,0,0,0,0,0,0,0 1,6,6,5,4,3,2,1
%e 18 45360 1,1,0,0,0,0,0,0 2,7,6,5,4,3,2,1
%e 19 80640 2,0,0,0,0,0,0,0 4,7,6,5,4,3,2,1
%e 20 120960 3,0,0,0,0,0,0,0 7,7,6,5,4,3,2,1
%e 21 362880 1,0,0,0,0,0,0,0,0 1,7,7,6,5,4,3,2,1
%e 22 403200 1,1,0,0,0,0,0,0,0 2,8,7,6,5,4,3,2,1
%e 23 725760 2,0,0,0,0,0,0,0,0 4,8,7,6,5,4,3,2,1
%e 24 1088640 3,0,0,0,0,0,0,0,0 7,8,7,6,5,4,3,2,1
%e 25 3628800 1,0,0,0,0,0,0,0,0,0 1,7,8,7,6,5,4,3,2,1
%e 26 3991680 1,1,0,0,0,0,0,0,0,0 2,8,8,7,6,5,4,3,2,1
%e 27 7257600 2,0,0,0,0,0,0,0,0,0 4,8,8,7,6,5,4,3,2,1
%e 28 7620480 2,1,0,0,0,0,0,0,0,0 6,9,8,7,6,5,4,3,2,1
%e 29 10886400 3,0,0,0,0,0,0,0,0,0 9,9,8,7,6,5,4,3,2,1
%e 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 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 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 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 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 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 Cf. A000142, A007623, A033312, A278742, A278743.
%K nonn,base
%O 1,2
%A _Rémy Sigrist_, Dec 18 2016
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