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A348138 For any number n with factorial base representation (d_1, ..., d_k), for i = 1..k, let f_i be the number of i's in (d_1, ..., d_k); the factorial base representation of a(n) is (f_1, ..., f_k).

%I A348138 #10 Oct 09 2021 06:30:30
%S A348138 0,1,2,4,1,3,6,12,12,18,8,14,2,8,8,14,4,10,1,7,7,13,3,9,24,48,48,72,
%T A348138 30,54,48,72,72,96,54,78,30,54,54,78,36,60,26,50,50,74,32,56,6,30,30,
%U A348138 54,12,36,30,54,54,78,36,60,12,36,36,60,18,42,8,32,32,56
%N A348138 For any number n with factorial base representation (d_1, ..., d_k), for i = 1..k, let f_i be the number of i's in (d_1, ..., d_k); the factorial base representation of a(n) is (f_1, ..., f_k).
%C A348138 Leading zeros in factorial base representation of n are ignored.
%H A348138 Rémy Sigrist, <a href="/A348138/b348138.txt">Table of n, a(n) for n = 0..5040</a>
%H A348138 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%F A348138 a(n) = n iff n = 0 or n is a factorial number (A000142).
%F A348138 A034968(a(n)) = A060130(n).
%F A348138 a(A153880(n)) = A153880(a(n)).
%e A348138 The first terms, in decimal and in factorial base, are:
%e A348138   n   a(n)  f(n)  f(a(n))
%e A348138   --  ----  ----  -------
%e A348138    0     0     0        0
%e A348138    1     1     1        1
%e A348138    2     2    10       10
%e A348138    3     4    11       20
%e A348138    4     1    20        1
%e A348138    5     3    21       11
%e A348138    6     6   100      100
%e A348138    7    12   101      200
%e A348138    8    12   110      200
%e A348138    9    18   111      300
%e A348138   10     8   120      110
%o A348138 (PARI) a(n) = { my (f=[]); for (r=2, oo, if (n==0, return (sum(k=1, #f, f[k]*(#f-k+1)!)), f=concat(f, 0); my (d=n%r); n\=r; if (d, f[d]++))) }
%Y A348138 Cf. A000142, A108731, A034968, A060130, A153880.
%K A348138 nonn,base
%O A348138 0,3
%A A348138 _Rémy Sigrist_, Oct 02 2021