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A263130 Least number such that the product of its digits in factorial base is n.

%I A263130 #14 Oct 27 2015 21:23:18
%S A263130 1,5,21,17,633,23,36153,65,93,635,443122713,71,81474226713,36155,645,
%T A263130 113,6069010670156313,95,2318037293294156313,641,36165,443122715,
%U A263130 595774037991797891660313,119,4233,81474226715,453,36161,256727294482662730300616548940313,647
%N A263130 Least number such that the product of its digits in factorial base is n.
%C A263130 The product of digits in factorial base is given by A208575.
%C A263130 All terms are odd.
%C A263130 Each prime number sets a new record.
%C A263130 a(p) = p*(p!) + Sum_{k=1..p-1} k! for any prime p.
%C A263130 a(n!) = A033312(n+1) for any n>0.
%C A263130 A208576(a(n)) = A208576(n)+1 for any n>1.
%H A263130 Paul Tek, <a href="/A263130/b263130.txt">Table of n, a(n) for n = 1..448</a>
%H A263130 Paul Tek, <a href="/A263130/a263130.pl.txt">PERL program for this sequence</a>
%e A263130 The first terms of the sequence are:
%e A263130 +----+-------------+----------------------------+
%e A263130 | n  | a(n)        | a(n) in factorial base     |
%e A263130 +----+-------------+----------------------------+
%e A263130 |  1 |           1 |                          1 |
%e A263130 |  2 |           5 |                        2_1 |
%e A263130 |  3 |          21 |                      3_1_1 |
%e A263130 |  4 |          17 |                      2_2_1 |
%e A263130 |  5 |         633 |                  5_1_1_1_1 |
%e A263130 |  6 |          23 |                      3_2_1 |
%e A263130 |  7 |       36153 |              7_1_1_1_1_1_1 |
%e A263130 |  8 |          65 |                    2_2_2_1 |
%e A263130 |  9 |          93 |                    3_3_1_1 |
%e A263130 | 10 |         635 |                  5_1_1_2_1 |
%e A263130 | 11 |   443122713 |     11_1_1_1_1_1_1_1_1_1_1 |
%e A263130 | 12 |          71 |                    2_3_2_1 |
%e A263130 | 13 | 81474226713 | 13_1_1_1_1_1_1_1_1_1_1_1_1 |
%e A263130 | 14 |       36155 |              7_1_1_1_1_2_1 |
%e A263130 | 15 |         645 |                  5_1_3_1_1 |
%e A263130 | 16 |         113 |                    4_2_2_1 |
%e A263130 +----+-------------+----------------------------+
%t A263130 f[n_] := Block[{d = Divisors@ n, g, k, m = {1}}, g[x_] := Flatten[Table[#1, {#2}] & @@@ FactorInteger@ x]; Do[k = Max@ Select[d, # <= i &]; If[! IntegerQ@ k, AppendTo[m, 1], d = Divisors[Last[d]/k]; AppendTo[m, k]]; If[d == {1}, Break[]], {i, 2, n}]; Reverse@ m]; Table[FromDigits[#, MixedRadix[Reverse@ Range[2, Length@ #]]] &@ f@ n, {n, 30}] (* _Michael De Vlieger_, Oct 12 2015, Version 10.2 *)
%Y A263130 Cf. A033312, A200748, A208277, A208575, A208576.
%K A263130 nonn,base
%O A263130 1,2
%A A263130 _Paul Tek_, Oct 10 2015