This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
A099564 begins {1,1,1,2,2,1,1,1,1,1,1,2,2,2,2,2,2,3,...}, so the first few records occur at a(1)=1, a(2)=4 and a(3)=18.
For n=15, f(15,2) = floor(15/2)=7, f(7,3)=2, f(2,4)=0, so a(15)=2. From _Antti Karttunen_, Dec 24 2015: (Start) 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)
Table[Floor[n/#] &@ (k = 1; While[(k + 1)! <= n, k++]; k!), {n, 0, 120}] (* Michael De Vlieger, Aug 30 2016 *)
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
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
(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 ;; Antti Karttunen, Dec 24-25 2015
For n=24, which is "400" in primorial base (as 24 = 4*(3*2*1) + 0*(2*1) + 0*1, see A049345), the most significant digit is 4, thus a(24) = 4. For n=210, which is "10000" in primorial base (as 210 = A002110(4) = 7*5*3*2*1), the most significant digit is 1, thus a(210) = 1. For n=2100, which could be written "A0000" in primorial base (where A stands for digit "ten", as 2100 = 10*A002110(4)), the most significant value holder is thus 10 and a(2100) = 10. (The first point where this sequence attains a value larger than 9).
nn = 120; Table[First@ IntegerDigits[n, MixedRadix[Reverse@ Prime@ Range@ PrimePi@ nn]], {n, 0, nn}] (* Michael De Vlieger, Aug 25 2016, Version 10.2 *)
(define (A276153 n) (let loop ((n n) (i 1)) (let* ((p (A000040 i)) (dig (modulo n p)) (next-n (/ (- n dig) p))) (if (zero? next-n) dig (loop next-n (+ 1 i))))))
In A097801-base, where the digit-positions are given by 1 and the terms of A097801 from its term a(1) onward: 2, 6, 30, 210, 1890, 20790, 270270, 4054050, ..., number 29 is expressed as "421" as 29 = 4*6 + 2*2 + 1*1, thus a(29) = 4. In the same base, number 30 is expressed as "1000" as 30 = 1*30, thus a(30) = 1. Number 1890 = 2*3*5*7*9 is expressed as "100000", thus a(1890) = 1.
Block[{nn = 105, b}, b = MixedRadix@ NestWhile[Prepend[#1, 2 #2 - 1] & @@ {#, Length[#] + 1} &, {2}, Times @@ # < nn &]; Array[First@ IntegerDigits[#, b] &, nn + 1, 0]] (* Michael De Vlieger, Feb 23 2021 *)
A341356(n) = { my(m=2, k=3); while(n>=m, n \= m; m = k; k += 2); (n); }; \\ Antti Karttunen & Kevin Ryde, Feb 24 2021
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