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A099563
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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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34
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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, 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, 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
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listen;
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internal format)
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OFFSET
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0,5
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COMMENTS
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Records in {a(n)} occur at {1,4,18,96,600,4320,35280,322560,3265920,...}, which appears to be n*n! = A001563(n).
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.
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LINKS
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FORMULA
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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)).
Other identities. For all n >= 0:
(End)
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EXAMPLE
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For n=15, f(15,2) = floor(15/2)=7, f(7,3)=2, f(2,4)=0, so a(15)=2.
Example illustrating the role of this sequence in factorial base representation:
n A007623(n) a(n) [= the most significant digit].
0 = 0 0
1 = 1 1
2 = 10 1
3 = 11 1
4 = 20 2
5 = 21 2
6 = 100 1
7 = 101 1
8 = 110 1
9 = 111 1
10 = 120 1
11 = 121 1
12 = 200 2
13 = 201 2
14 = 210 2
15 = 211 2
16 = 220 2
17 = 221 2
18 = 300 3
etc.
Note that there is no any upper bound for the size of digits in this representation.
(End)
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MATHEMATICA
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Table[Floor[n/#] &@ (k = 1; While[(k + 1)! <= n, k++]; k!), {n, 0, 120}] (* Michael De Vlieger, Aug 30 2016 *)
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PROG
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(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
(Scheme)
(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))))))
(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
(Python)
def a(n):
i=2
d=0
while n:
d=n%i
n=(n - d)//i
i+=1
return d
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 21 2017, after PARI code
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CROSSREFS
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Cf. also A034968, A048764, A051683, A055881, A126307, A230420, A246359, A249069, A257679, A257684, A257686, A257687, A265890, A265891, A265894, A265333, A265334.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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a(0) = 0 prepended and the alternative description added to the name-field by Antti Karttunen, Dec 24 2015
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STATUS
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approved
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