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.
a(12) = 4 because 1, 2, 3, 4 divide 12, but 5 does not.
a055874 n = length $ takeWhile ((== 0) . (mod n)) [1..] -- Reinhard Zumkeller, Feb 21 2012, Dec 09 2010
N:= 1000: # to get a(1) to a(N)
A:= Vector(N,1);
for m from 2 do
Lm:= ilcm($1..m);
if Lm > N then break fi;
if Lm mod (m+1) = 0 then next fi;
for k from 1 to floor(N/Lm) do
A[k*Lm]:=m
od
od:
convert(A,list); # Robert Israel, Nov 28 2014
a[n_] := Module[{m = 1}, While[Divisible[n, m++]]; m - 2]; Array[a, 100] (* Jean-François Alcover, Mar 07 2016 *)
a(n) = my(m = 1); while ((n % m) == 0, m++); m - 1; \\ Michel Marcus, Jan 17 2014
from itertools import count def A055874(n): for m in count(1): if n % m: return m-1 # Chai Wah Wu, Jan 02 2022
(define (A055874 n) (let loop ((m 1)) (if (not (zero? (modulo n m))) (- m 1) (loop (+ 1 m))))) ;; Antti Karttunen, Nov 18 2013
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))))))
Module[{nn=10,fct},fct=Table[{f,f!},{f,nn}];Table[Select[fct,Mod[n^2,#[[2]]]==0&][[-1,1]],{n,90}]] (* Harvey P. Dale, Aug 11 2024 *)
(define (A232098 n) (A055881 (A000290 n)))
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.
210 (= 2*3*5*7) is a member, because A053669(210)=11, while A236454(210) = A007978(210*210) = A007978(44100) = 8. 446185740 (= 2*2*3*5*7*11*13*17*19*23) is a member, because A053669(446185740) = 29, while A236454(446185740) = 27, as there is only one 3 present in 446185740, which means that its square is only divisible by 9, but not by 27 = 3^3.
840 (= 3*5*7*8) is in the sequence as all natural numbers up to 8 divide 840, but the largest factorial that divides its square, 705600, is 7! (840^2 = 140 * 5040), and 7 differs from 8.
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.
a[n_] := Module[{m = 1}, While[Divisible[n^2, m++]]; m - 2]; Array[a, 100] (* Jean-François Alcover, Mar 07 2016 *)
a(n) = my(m = 1); while ((n^2 % m) == 0, m++); m - 1; \\ Michel Marcus, Jan 17 2014
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