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a(0) = 0; for n > 0, a(n) = final nonzero number in the sequence n, f(n,2), f(f(n,2),3), f(f(f(n,2),3),4),..., where f(n,d) = floor(n/d); the most significant digit in the factorial base representation of n.
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%I #33 Mar 10 2021 07:23:26

%S 0,1,1,1,2,2,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,

%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,

%U 2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4

%N a(0) = 0; for n > 0, a(n) = final nonzero number in the sequence n, f(n,2), f(f(n,2),3), f(f(f(n,2),3),4),..., where f(n,d) = floor(n/d); the most significant digit in the factorial base representation of n.

%C Records in {a(n)} occur at {1,4,18,96,600,4320,35280,322560,3265920,...}, which appears to be n*n! = A001563(n).

%C The most significant digit in the factorial expansion of n (A007623). Proof: The algorithm that computes the factorial expansion of n, generates the successive digits by repeatedly dividing the previous quotient with successively larger divisors (the remainders give the digits), starting from n itself and divisor 2. As a corollary we find that A001563 indeed gives the positions of the records. - _Antti Karttunen_, Jan 01 2007.

%H Antti Karttunen, <a href="/A099563/b099563.txt">Table of n, a(n) for n = 0..10080</a>

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

%F From _Antti Karttunen_, Dec 25 2015: (Start)

%F a(0) = 0; for n >= 1, if A265333(n) = 1 [when n is one of the terms of A265334], a(n) = 1, otherwise 1 + a(A257684(n)).

%F Other identities. For all n >= 0:

%F a(A001563(n)) = n. [Sequence works as a left inverse for A001563.]

%F a(n) = A257686(n) / A048764(n).

%F (End)

%e For n=15, f(15,2) = floor(15/2)=7, f(7,3)=2, f(2,4)=0, so a(15)=2.

%e From _Antti Karttunen_, Dec 24 2015: (Start)

%e Example illustrating the role of this sequence in factorial base representation:

%e n A007623(n) a(n) [= the most significant digit].

%e 0 = 0 0

%e 1 = 1 1

%e 2 = 10 1

%e 3 = 11 1

%e 4 = 20 2

%e 5 = 21 2

%e 6 = 100 1

%e 7 = 101 1

%e 8 = 110 1

%e 9 = 111 1

%e 10 = 120 1

%e 11 = 121 1

%e 12 = 200 2

%e 13 = 201 2

%e 14 = 210 2

%e 15 = 211 2

%e 16 = 220 2

%e 17 = 221 2

%e 18 = 300 3

%e etc.

%e Note that there is no any upper bound for the size of digits in this representation.

%e (End)

%t Table[Floor[n/#] &@ (k = 1; While[(k + 1)! <= n, k++]; k!), {n, 0, 120}] (* _Michael De Vlieger_, Aug 30 2016 *)

%o (PARI) A099563(n) = { my(i=2,dig=0); until(0==n, dig = n % i; n = (n - dig)/i; i++); return(dig); }; \\ _Antti Karttunen_, Dec 24 2015

%o (Scheme)

%o (define (A099563 n) (let loop ((n n) (i 2)) (let* ((dig (modulo n i)) (next-n (/ (- n dig) i))) (if (zero? next-n) dig (loop next-n (+ 1 i))))))

%o (definec (A099563 n) (cond ((zero? n) n) ((= 1 (A265333 n)) 1) (else (+ 1 (A099563 (A257684 n)))))) ;; Based on given recurrence, using the memoization-macro definec

%o ;; _Antti Karttunen_, Dec 24-25 2015

%o (Python)

%o def a(n):

%o i=2

%o d=0

%o while n:

%o d=n%i

%o n=(n - d)//i

%o i+=1

%o return d

%o print([a(n) for n in range(201)]) # _Indranil Ghosh_, Jun 21 2017, after PARI code

%Y Cf. A001563, A007623, A099564.

%Y Cf. also A034968, A048764, A051683, A055881, A126307, A230420, A246359, A249069, A257679, A257684, A257686, A257687, A265890, A265891, A265894, A265333, A265334.

%K nonn

%O 0,5

%A _John W. Layman_, Oct 22 2004

%E a(0) = 0 prepended and the alternative description added to the name-field by _Antti Karttunen_, Dec 24 2015