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.
The first rows of this triangular table are: 1; 1, 1; 1, 2, 2; 4, 4, 5, 6; 12, 14, 16, 19, 22; 55, 59, 65, 73, 82, 94; ...
nn = 10^3; m = 1; While[m! < Floor[6 nn/5], m++]; m; t = TakeWhile[Reverse@ NestWhileList[# - Total@ IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]] &, Floor[6 nn/5], # > 0 &], # <= nn &] (* Michael De Vlieger, Jun 27 2016, Version 10.2 *)
;; Memoizing definec-macro from Antti Karttunen's IntSeq-library (definec (A219666 n) (cond ((<= n 2) n) ((= (A226061 (A230411 n)) n) (- (A000142 (A230411 n)) 1)) (else (- (A219666 (+ n 1)) (A034968 (A219666 (+ n 1))))))) ;; Another variant, utilizing A230416 (which gives a more convenient way to compute large number of terms of this sequence): (define (A219666 n) (A230416 (A230432 n))) ;; This function is for checking whether n belongs to this sequence: (define (inA219666? n) (or (zero? n) (= 1 (- (A230418 (+ 1 n)) (A230418 n)))))
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
(1!)-1 (0) is reached from (2!)-1 (1) with one step by subtracting A034968(1) from 1. (2!)-1 (1) is reached from (3!)-1 (5) with two steps by first subtracting A034968(5) from 5 -> 2, and then subtracting A034968(2) from 2 -> 1. (3!)-1 (5) is reached from (4!)-1 (23) with five steps by repeatedly subtracting the sum of digits in factorial expansion as: 23 - 6 = 17, 17 - 5 = 12, 12 - 2 = 10, 10 - 3 = 7, 7 - 2 = 5. Thus a(1)=1, a(2)=2 and a(3)=5.
Table[Length@ NestWhileList[# - Total@ IntegerDigits[#, MixedRadix[Reverse@ Range[2, 120]]] &, (n + 1)! - 1, # > n! - 1 &] - 1, {n, 0, 8}] (* Michael De Vlieger, Jun 27 2016, Version 10.2 *)
(define (A219661 n) (if (zero? n) n (let loop ((i (-1+ (A000142 (1+ n)))) (steps 1)) (cond ((isA000142? (1+ (A219651 i))) steps) (else (loop (A219651 i) (1+ steps))))))) (define (isA000142? n) (and (> n 0) (let loop ((n n) (i 2)) (cond ((= 1 n) #t) ((not (zero? (modulo n i))) #f) (else (loop (/ n i) (1+ i))))))) ;; Alternative definition: (define (A219661 n) (- (A219652 (-1+ (A000142 (1+ n)))) (A219652 (-1+ (A000142 n)))))
The first rows of this triangular table are: 1; 2, 5; 7, 12, 23; 25, 48, 74, 97; 121, 240, 362, 481, 605; ... T(3,1) = 7 as 7 has factorial base representation 101, which is the smallest such three digit term in A219666 beginning with factorial base digit 1 (in other words, for which A084558(x)=3 and A099563(x)=1). T(3,2) = 12 as 12 has factorial base representation 200, which is the smallest such three digit term in A219666 beginning with factorial base digit 2. T(3,3) = 23 as 23 has factorial base representation 321, which is the smallest such three digit term in A219666 beginning with factorial base digit 3.
The first rows of this triangular table are: 1; 2, 5; 10, 17, 23; 46, 70, 92, 119; 238, 358, 476, 597, 719; ... T(3,1) = 10 as 10 has factorial base representation 120, which is the largest such three digit term in A219666 beginning with factorial base digit 1 (in other words, for which A084558(x)=3 and A099563(x)=1). T(3,2) = 17 as 17 has factorial base representation 221, which is the largest such three digit term in A219666 beginning with factorial base digit 2. T(3,3) = 23 as 23 has factorial base representation 321, which is the largest such three digit term in A219666 beginning with factorial base digit 3.
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