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A257687 Discard the most significant digit from factorial base representation of n, then convert back to decimal: a(n) = n - A257686(n).

%I A257687 #32 Mar 13 2021 12:43:39
%S A257687 0,0,0,1,0,1,0,1,2,3,4,5,0,1,2,3,4,5,0,1,2,3,4,5,0,1,2,3,4,5,6,7,8,9,
%T A257687 10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,
%U A257687 12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0
%N A257687 Discard the most significant digit from factorial base representation of n, then convert back to decimal: a(n) = n - A257686(n).
%C A257687 A060130(n) gives the number of steps needed to reach zero, when starting iterating as a(k), a(a(k)), etc., from the starting value k = n.
%H A257687 Antti Karttunen, <a href="/A257687/b257687.txt">Table of n, a(n) for n = 0..10080</a>
%F A257687 a(n) = n - A257686(n).
%e A257687 Factorial base representation (A007623) of 1 is "1", discarding the most significant digit leaves nothing, taken to be zero, thus a(1) = 0.
%e A257687 Factorial base representation of 2 is "10", discarding the most significant digit leaves "0", thus a(2) = 0.
%e A257687 Factorial base representation of 3 is "11", discarding the most significant digit leaves "1", thus a(3) = 1.
%e A257687 Factorial base representation of 4 is "20", discarding the most significant digit leaves "0", thus a(4) = 0.
%t A257687 f[n_] := Block[{m = p = 1}, While[p*(m + 1) <= n, p = p*m; m++]; Mod[n, p]]; Array[f, 101, 0] (* _Robert G. Wilson v_, Jul 21 2015 *)
%o A257687 (Scheme) (define (A257687 n) (- n (A257686 n)))
%o A257687 (Python)
%o A257687 from sympy import factorial as f
%o A257687 def a007623(n, p=2): return n if n<p else a007623(n//p, p+1)*10 + n%p
%o A257687 def a(n):
%o A257687     x=str(a007623(n))[1:][::-1]
%o A257687     return sum(int(x[i])*f(i + 1) for i in range(len(x)))
%o A257687 print([a(n) for n in range(201)]) # _Indranil Ghosh_, Jun 21 2017
%Y A257687 Cf. A007623, A257686.
%Y A257687 Can be used (together with A099563) to define simple recurrences for sequences like A034968, A060130, A227153, A246359, A257511, A257679, A257680.
%Y A257687 Cf. also A257684.
%K A257687 nonn,base
%O A257687 0,9
%A A257687 _Antti Karttunen_, May 04 2015