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.
In factorial base A007623, 0 is shown as "0", but in this case all the zeros are trailing, so we set a(0) = 0 by convention. For n = 2, A007623(2) = "10", and by discarding the trailing zero only one significant digit "1" is left, thus a(2) = 1. For n = 132, A007623(132) = "10200", and by discarding its trailing zeros we are left with just three digits "102", thus a(132) = 3.
a[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; Length[s] - FirstPosition[s, ?(#>0 &)][[1]] + 1]; a[0] = 0; Array[a, 100, 0] (* _Amiram Eldar, Feb 21 2024 *)
(define (A265893 n) (- (A084558 n) (A230403 n)))
a(12) = 3 because 3! is highest factorial to divide 12. From _Joerg Arndt_, Jul 16 2011: (Start) All permutations of 4 elements via prefix reversals: n: permutation a(n)+1 0: [ 0 1 2 3 ] - 1: [ 1 0 2 3 ] 2 2: [ 2 0 1 3 ] 3 3: [ 0 2 1 3 ] 2 4: [ 1 2 0 3 ] 3 5: [ 2 1 0 3 ] 2 6: [ 3 0 1 2 ] 4 7: [ 0 3 1 2 ] 2 8: [ 1 3 0 2 ] 3 9: [ 3 1 0 2 ] 2 10: [ 0 1 3 2 ] 3 11: [ 1 0 3 2 ] 2 12: [ 2 3 0 1 ] 4 13: [ 3 2 0 1 ] 2 14: [ 0 2 3 1 ] 3 15: [ 2 0 3 1 ] 2 16: [ 3 0 2 1 ] 3 17: [ 0 3 2 1 ] 2 18: [ 1 2 3 0 ] 4 19: [ 2 1 3 0 ] 2 20: [ 3 1 2 0 ] 3 21: [ 1 3 2 0 ] 2 22: [ 2 3 1 0 ] 3 23: [ 3 2 1 0 ] 2 (End) From _Joerg Arndt_, Dec 15 2012: (Start) The first few rising factorial numbers (dots for zeros) with 4 digits and the positions of the rightmost change with incrementing are: [ 0] [ . . . . ] - [ 1] [ 1 . . . ] 1 [ 2] [ . 1 . . ] 2 [ 3] [ 1 1 . . ] 1 [ 4] [ . 2 . . ] 2 [ 5] [ 1 2 . . ] 1 [ 6] [ . . 1 . ] 3 [ 7] [ 1 . 1 . ] 1 [ 8] [ . 1 1 . ] 2 [ 9] [ 1 1 1 . ] 1 [10] [ . 2 1 . ] 2 [11] [ 1 2 1 . ] 1 [12] [ . . 2 . ] 3 [13] [ 1 . 2 . ] 1 [14] [ . 1 2 . ] 2 [15] [ 1 1 2 . ] 1 [16] [ . 2 2 . ] 2 [17] [ 1 2 2 . ] 1 [18] [ . . 3 . ] 3 [19] [ 1 . 3 . ] 1 [20] [ . 1 3 . ] 2 [21] [ 1 1 3 . ] 1 [22] [ . 2 3 . ] 2 [23] [ 1 2 3 . ] 1 [24] [ . . . 1 ] 4 [25] [ 1 . . 1 ] 1 [26] [ . 1 . 1 ] 2 (End)
Table[Length[Intersection[Divisors[n], Range[5]!]], {n, 125}] (* Alonso del Arte, Dec 10 2012 *)
f[n_] := Block[{m = 1}, While[Mod[n, m!] == 0, m++]; m - 1]; Array[f, 105] (* Robert G. Wilson v, Dec 21 2012 *)
See Cano link.
n=5; f=n!; x='x+O('x^f); Vec(sum(k=1,n,x^(k!)/(1-x^(k!)))) \\ Joerg Arndt, Jan 28 2014
a(n)=for(k=2,n+1,if(n%k, return(k-1),n/=k)) \\ Charles R Greathouse IV, May 28 2015
(define (A055881 n) (let loop ((n n) (i 2)) (cond ((not (zero? (modulo n i))) (- i 1)) (else (loop (/ n i) (+ 1 i))))))
For n=24, which is "400" in primorial base (as 24 = 4*(3*2*1) + 0*(2*1) + 0*1, see A049345), there are two trailing zeros, thus a(24) = 2.
Table[If[# == 0, 0, j = #; While[! Divisible[n, Times @@ Prime@ Range@ j], j--]; j] &@ If[OddQ@ n, 0, k = 1; While[Times @@ Prime@ Range[k + 1] <= n, k++]; k], {n, 120}] (* or *)
nn = 120; b = MixedRadix[Reverse@ Prime@ Range@ PrimePi[nn + 1]]; Table[Length@ TakeWhile[Reverse@ IntegerDigits[n, b], # == 0 &], {n, nn}] (* Version 10.2, or *)
f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Prime@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Prime@ Range[# + 1] <= n &]; Rest[a][[All, 1]]]; Table[Length@ TakeWhile[Reverse@ f@ n, # == 0 &], {n, 120}] (* Michael De Vlieger, Aug 30 2016 *)
from sympy import nextprime, primepi
def a053669(n):
p = 2
while True:
if n%p!=0: return p
else: p=nextprime(p)
def a(n): return primepi(a053669(n)) - 1 # Indranil Ghosh, May 12 2017
(define (A276084 n) (let loop ((n n) (i 1)) (let* ((p (A000040 i)) (d (modulo n p))) (if (not (zero? d)) (- i 1) (loop (/ (- n d) p) (+ 1 i))))))
factBaseIntDs[n_] := Module[{m, i, len, dList, currDigit}, i = 1; While[n > i!, i++]; m = n; len = i; dList = Table[0, {len}]; Do[currDigit = 0; While[m >= j!, m = m - j!; currDigit++]; dList[[len - j + 1]] = currDigit, {j, i, 1, -1}]; If[dList[[1]] == 0, dList = Drop[dList, 1]]; dList]; s = Table[FromDigits[factBaseIntDs[n]], {n, 120}]; Last@ DigitCount[#] & /@ s (* Michael De Vlieger, Apr 27 2015, after Alonso del Arte at A007623 *)
(define (A257510 n) (let loop ((n n) (i 2) (s 0)) (cond ((zero? n) s) (else (loop (floor->exact (/ n i)) (+ 1 i) (+ s (if (zero? (modulo n i)) 1 0)))))))
120 is included because A055881(120)=5 and 5+1 is a composite number larger than 5. Note that A007623(120) = '10000', with four trailing zeros. 720 is the first missing multiple of 120, as A055881(720)=6 and 7 is a prime, not composite, so 720 is not included in this sequence. Note that A007623(720) = '100000', with five trailing zeros, and 5+2 is not a composite. 120960 (= 3*8!) is included because A055881(120960)=8 and 9 is a composite number larger than 5. Note that A007623(120960) = '30000000', with seven trailing zeros.
6 is included because A055881(6)=3 and 3+1 is a composite number. 24 is the first excluded multiple of 6, as A055881(24)=4 and 5 is a prime, not composite, so 24 is not included in this sequence. 120 is the first included multiple of 24, as A055881(120)=5 and 6 is a composite.
For n = 1, with factorial base representation (A007623) "1", there are no nonleading zeros at all, thus a(1) = 0. For n = 6, with representation "100", the rightmost zero occurs at digit-position 1 (when the least significant digit has index 1, etc.), thus a(6) = 1. For n = 7, with representation "101", the rightmost zero occurs at position 2, thus a(7) = 2.
a[n_] := Module[{k = n, m = 2, r, s = {}, p}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; If[MissingQ[(p = FirstPosition[s, 0])], 0, p[[1]]]]; Array[a, 100] (* Amiram Eldar, Feb 07 2024 *)
(define (A257260 n) (let loop ((n n) (i 2)) (cond ((zero? n) 0) ((zero? (modulo n i)) (- i 1)) (else (loop (floor->exact (/ n i)) (+ 1 i))))))
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