A299199 - OEIS

%I #12 Feb 14 2018 03:37:50

%S 1,4,14,98,2,4386,18,324,60,36457092,12,5769254382,2598,78,414,

%T 335391687123174,510,115428139222691670,30,1926,20204166,

%U 24752828962220504429646,6,1032336,3124309416,149376,3816,8542182056001396008878674488976,96

%N In factorial base, any rational number has a terminating expansion; hence we can devise a self-inverse permutation of the rational numbers, say f, such that for any rational number q, the representations of q and of f(q) in factorial base are mirrored around the radix point and q and f(q) have the same sign; a(n) = f(1/n).

%C See A299161 for additional comments about f.

%C This sequence corresponds to the indices of ones in A299161.

%H Rémy Sigrist, <a href="/A299199/b299199.txt">Table of n, a(n) for n = 2..456</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Factorial_number_system#Fractional_values">Factorial number system (Fractional values)</a>

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>

%H Rémy Sigrist, <a href="/A299199/a299199.png">Colored logarithmic scatterplot of the first 5000 terms</a> (where the color is function of A299160(n))

%F A034968(a(n)) = A276350(n) for any n > 1.

%F A299160(a(n)) = n for any n > 1.

%F A299161(a(n)) = 1 for any n > 1.

%e The first terms, alongside the factorial base representations of a(n) and of 1/n, are:

%e   n    a(n)         fact(a(n))                fact(1/n)

%e   --   ----------   -----------------------   ------------

%e    2            1                         1   0.1

%e    3            4                       2 0   0.0 2

%e    4           14                     2 1 0   0.0 1 2

%e    5           98                   4 0 1 0   0.0 1 0 4

%e    6            2                       1 0   0.0 1

%e    7         4386               6 0 2 3 0 0   0.0 0 3 2 0 6

%e    8           18                     3 0 0   0.0 0 3

%e    9          324                 2 3 2 0 0   0.0 0 2 3 2

%e   10           60                   2 2 0 0   0.0 0 2 2

%e   11     36457092      10 0 4 1 3 5 0 2 0 0   0.0 0 2 0 5 3 1 4 0 10

%e   12           12                     2 0 0   0.0 0 2

%e   13   5769254382  12 0 5 8 4 5 2 1 4 1 0 0   0.0 0 1 4 1 2 5 4 8 5 0 12

%e   14         2598               3 3 3 1 0 0   0.0 0 1 3 3 3

%e   15           78                   3 1 0 0   0.0 0 1 3

%e   16          414                 3 2 1 0 0   0.0 0 1 2 3

%o (PARI) a(n) = my (v=0, q=1/n); for (r=2, oo, q *= r; v += floor(q) * (r-1)!; q = frac(q); if (q==0, return (v)))

%Y Cf. A034968, A052126, A276350, A299160, A299161.

%K nonn,base

%O 2,2

%A _Rémy Sigrist_, Feb 04 2018