A055881 -id:A055881 - cp's OEIS frontend

A055874 a(n) = largest m such that 1, 2, ..., m divide n.

1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2

Offset: 1

Leroy Quet, Jul 16 2000

From Antti Karttunen, Nov 20 2013 & Jan 26 2014: (Start)
Differs from A232098 for the first time at n=840, where a(840)=8, while A232098(840)=7. A232099 gives all the differing positions. See also the comments at A055926 and A232099.
The positions where a(n) is an odd prime is given by A017593 up to A017593(34)=414 (so far all 3's), after which comes the first 7 at a(420). (A017593 gives the positions of 3's.)
(Continued on Jan 26 2014):
Only terms of A181062 occur as values.
A235921 gives such n where a(n^2) (= A235918(n)) differs from A071222(n-1) (= A053669(n)-1). (End)
a(n) is the largest m such that A003418(m) divides n. - David W. Wilson, Nov 20 2014
a(n) is the largest number of consecutive integers dividing n. - David W. Wilson, Nov 20 2014
A051451 gives indices where record values occur. - Gionata Neri, Oct 17 2015
Yuri Matiyasevich calls this the maximum inheritable divisor of n. - N. J. A. Sloane, Dec 14 2023

			a(12) = 4 because 1, 2, 3, 4 divide 12, but 5 does not.

Antti Karttunen, Table of n, a(n) for n = 1..10080
Bakir Farhi, On the average asymptotic behavior of a certain type of sequences of integers, Integers, Vol. 9 (2009), pp. 555-567.

One less than A007978.

Cf. also A053669, A055881, A055926, A017593, A064859, A181062, A126800, A232098, A232099, A233284, A235918, A235921.

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(n) = A007978(n) - 1. - Antti Karttunen, Jan 26 2014

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A064859 (Farhi, 2009). - Amiram Eldar, Jul 25 2022

A099563 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.

Original entry on oeis.org

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

Offset: 0

John W. Layman, Oct 22 2004

nonn

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.

			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)

Antti Karttunen, Table of n, a(n) for n = 0..10080
Index entries for sequences related to factorial base representation

Cf. A001563, A007623, A099564.

Cf. also A034968, A048764, A051683, A055881, A126307, A230420, A246359, A249069, A257679, A257684, A257686, A257687, A265890, A265891, A265894, A265333, A265334.

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

From Antti Karttunen, Dec 25 2015: (Start)
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:
a(A001563(n)) = n. [Sequence works as a left inverse for A001563.]
a(n) = A257686(n) / A048764(n).
(End)

a(0) = 0 prepended and the alternative description added to the name-field by Antti Karttunen, Dec 24 2015

A076934 Smallest integer of the form n/k!.

Original entry on oeis.org

1, 1, 3, 2, 5, 1, 7, 4, 9, 5, 11, 2, 13, 7, 15, 8, 17, 3, 19, 10, 21, 11, 23, 1, 25, 13, 27, 14, 29, 5, 31, 16, 33, 17, 35, 6, 37, 19, 39, 20, 41, 7, 43, 22, 45, 23, 47, 2, 49, 25, 51, 26, 53, 9, 55, 28, 57, 29, 59, 10, 61, 31, 63, 32, 65, 11, 67, 34, 69, 35, 71

Offset: 1

Amarnath Murthy, Oct 19 2002

Equivalently, n divided by the largest factorial divisor of n.
Also, the smallest r such that n/r is a factorial number.
Positions of 1's are the factorial numbers A000142. Is every positive integer in this sequence? - Gus Wiseman, May 15 2019
Let m = A055874(n), the largest integer such that 1,2,...,m divides n. Then a(n*m!) = n since m+1 does not divide n, showing that every integer is part of the sequence. - Etienne Dupuis, Sep 19 2020

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000142, A011371, A022559, A055874, A055881, A071626, A111701, A115627, A135291, A229020.

Cf. A325272, A325273, A325508, A325509, A325543, A325544.

Table[n/Max@@Intersection[Divisors[n],Array[Factorial,n]],{n,100}] (* Gus Wiseman, May 15 2019 *)
a[n_] := Module[{k=1}, While[Divisible[n, k!], k++]; n/(k-1)!]; Array[a, 100] (* Amiram Eldar, Dec 25 2023 *)

first(n) = {my(res = [1..n]); for(i = 2, oo, k = i!; if(k <= n, for(j = 1, n\k, res[j*k] = j ) , return(res) ) ) } \\ David A. Corneth, Sep 19 2020

From Amiram Eldar, Dec 25 2023: (Start)
a(n) = n/A055881(n)!.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = BesselI(2, 2) = 0.688948... (A229020). (End)

More terms from David A. Corneth, Sep 19 2020

A230403 a(n) = the largest k such that (k+1)! divides n; the number of trailing zeros in the factorial base representation of n (A007623(n)).

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1

Offset: 1

Antti Karttunen, Oct 31 2013

Many of the comments given in A055881 apply also here.
From Amiram Eldar, Mar 10 2021: (Start)
The asymptotic density of the occurrences of k is (k+1)/(k+2)!.
The asymptotic mean of this sequence is e - 2 = 0.718281... (A001113 - 2). (End)

			In factorial number base representation (A007623), the numbers from 1 to 9 are represented as:
n  A007623(n)  a(n) (gives the number of trailing zeros)
1        1       0
2       10       1
3       11       0
4       20       1
5       21       0
6      100       2
7      101       0
8      110       1
9      111       0

Antti Karttunen, Table of n, a(n) for n = 1..10080
Tyler Ball, Joanne Beckford, Paul Dalenberg, Tom Edgar, and Tina Rajabi, Some Combinatorics of Factorial Base Representations, J. Int. Seq., Vol. 23 (2020), Article 20.3.3.
Index entries for sequences related to factorial base representation

Cf. A001113, A055881. Bisection: A230404.
A few sequences related to factorial base representation (A007623): A034968, A084558, A099563, A060130, A227130, A227132, A227148, A227149, A153880.
Analogous sequence for binary system: A007814.

With[{b = MixedRadix[Range[12, 2, -1]]}, Array[LengthWhile[Reverse@ IntegerDigits[#, b], # == 0 &] &, 105]] (* Michael De Vlieger, Jun 03 2020 *)

(define (A230403 n) (if (zero? n) 0 (let loop ((n n) (i 2)) (cond ((not (zero? (modulo n i))) (- i 2)) (else (loop (/ n i) (1+ i)))))))

a(n) = A055881(n)-1.

A055926 Numbers k such that {largest m such that 1, 2, ..., m divide k} is different from {largest m such that m! divides k}; numbers k which are either odd multiples of 12 or the largest m such that (m-1)! divides k is a composite number > 5.

Original entry on oeis.org

12, 36, 60, 84, 108, 120, 132, 156, 180, 204, 228, 240, 252, 276, 300, 324, 348, 360, 372, 396, 420, 444, 468, 480, 492, 516, 540, 564, 588, 600, 612, 636, 660, 684, 708, 732, 756, 780, 804, 828, 840, 852, 876, 900, 924, 948, 960, 972, 996, 1020, 1044, 1068

Offset: 1

Leroy Quet, Jul 16 2000

From Antti Karttunen, Nov 20 - Dec 06 2013: (Start)
This sequence has several interpretations:
Numbers k such that A055874(k) differs from A055881(k). [Leroy Quet's original definition of the sequence. Note that A055874(k) >= A055881(k) for all k.]
Numbers k such that {largest m such that m! divides k^2} is different from {largest m such that m! divides k}, i.e., numbers k for which A232098(k) > A055881(k).
Numbers k which are either 12 times an odd number (A073762) or the largest m such that (m-1)! divides k is a composite number > 5 (A232743).
Please see my attached notes for the proof of the equivalence of these interpretations.
Additional implications based on that proof:
A232099 is a subset of this sequence.
A055881(a(n))+1 is always composite. In the range n = 1..17712, only values 4, 6, 8, 9 and 10 occur.
The new definition can be also rephrased by saying that the sequence contains all the positive integers k whose factorial base representation of (A007623(k)) either ends as '...200' (in which case k is an odd multiple of 12, 12 = '200', 36 = '1200', 60 = '2200', ...) or the number of trailing zeros + 2 in that representation is a composite number greater than or equal to 6, e.g. 120 = '10000' (in other words, A055881(k) is one of the terms of A072668 after the initial 3). Together these conditions also imply that all the terms are divisible by 12.
(End)

			12 is included because 3! is the largest factorial to divide 12, but 1, 2, 3 and 4 all divide 12. Equally, 12 is included because it is one of the terms of A073762, or equally, because its factorial base representation ends with digits '...200': A007623(12) = 200.
840 (= 3*5*7*8) is included because the largest factorial which divides 840 is 5! (840 = 7*120), but all positive integers up to 8 divide 840. Equally, 840 is included because it is one of the terms of A232743 as 5+1 = 6 is a composite number larger than 5. Note that A007623(840) = 110000.

Antti Karttunen, Table of n, a(n) for n = 1..17712
Antti Karttunen, A proof for the equivalence of three alternative definitions of A055926
Wikipedia, Wilson's theorem (See the section "Composite modulus")

Union of A073762 and A232743. Equivalently, setwise difference of A232742 and A017593. Subset: A232099.

Cf. A055874, A055881, A072668, A232098, A232100, A232741, A232744, A232745.

More terms from Antti Karttunen, Dec 01 2013

A232745 Numbers k for which the largest m such that m! divides k is even.

Original entry on oeis.org

2, 4, 8, 10, 14, 16, 20, 22, 24, 26, 28, 32, 34, 38, 40, 44, 46, 48, 50, 52, 56, 58, 62, 64, 68, 70, 72, 74, 76, 80, 82, 86, 88, 92, 94, 96, 98, 100, 104, 106, 110, 112, 116, 118, 122, 124, 128, 130, 134, 136, 140, 142, 144, 146, 148, 152, 154, 158, 160, 164

Offset: 1

Antti Karttunen, Dec 01 2013

nonn

Numbers k for which A055881(k) is even.
Equally: Numbers k which have an odd number of the trailing zeros in their factorial base representation A007623(k).
The sequence can be described in the following manner: Sequence includes all multiples of 2! (even numbers), except that it excludes from those the multiples of 3! (6), except that it includes the multiples of 4! (24), except that it excludes the multiples of 5! (120), except that it includes the multiples of 6! (720), except that it excludes the multiples of 7! (5040), except that it includes the multiples of 8! (40320), except that it excludes the multiples of 9! (362880), except that it includes the multiples of 10! (3628800), except that ..., ad infinitum.
The number of terms not exceeding m! for m>=1 is A000166(m). The asymptotic density of this sequence is 1/e (A068985). - Amiram Eldar, Feb 26 2021

Antti Karttunen, Table of n, a(n) for n = 1..14833

Complement: A232744.
b(n) = A153880(A232744(n)) gives a subset of this sequence.
Cf. also A055881, A007623, A232741-A232743.
Analogous sequences for binary system: A003159 & A036554.
Cf. A000166, A068985.

seq[max_] := Select[Range[max!], OddQ @ LengthWhile[Reverse @ IntegerDigits[#, MixedRadix[Range[max, 2, -1]]], #1 == 0 &] &]; seq[5] (* Amiram Eldar, Feb 26 2021 *)

A232744 Numbers k for which the largest m such that m! divides k is odd.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 25, 27, 29, 30, 31, 33, 35, 36, 37, 39, 41, 42, 43, 45, 47, 49, 51, 53, 54, 55, 57, 59, 60, 61, 63, 65, 66, 67, 69, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 97, 99, 101, 102, 103, 105

Offset: 1

Antti Karttunen, Dec 01 2013

nonn

Numbers k for which A055881(k) is odd.
Equally: Numbers k which have an even number of the trailing zeros in their factorial base representation A007623(k).
The sequence can be described in the following manner: Sequence includes all multiples of 1!, except that it excludes from those the multiples of 2!, except that it includes the multiples of 3! (6), except that it excludes the multiples of 4! (24), except that it includes the multiples of 5! (120), except that it excludes the multiples of 6! (720), except that it includes the multiples of 7! (5040), except that it excludes the multiples of 8! (40320), except that it includes the multiples of 9! (362880), and so on, ad infinitum.
The number of terms not exceeding m! for m>=1 is A002467(m). The asymptotic density of this sequence is 1 - 1/e (A068996). - Amiram Eldar, Feb 26 2021

Antti Karttunen, Table of n, a(n) for n = 1..9558

Complement: A232745. Cf. also A055881, A007623, A232741-A232743.
Analogous sequences for binary system: A003159 & A036554.
Cf. A002467, A068996.

seq[max_] := Select[Range[max!], EvenQ @ LengthWhile[Reverse @ IntegerDigits[#, MixedRadix[Range[max, 2, -1]]], #1 == 0 &] &]; seq[5] (* Amiram Eldar, Feb 26 2021 *)

a(1)=1, and for n>1, a(n) = a(n-1) + (2 - A000035(A055881(a(n-1)+1))).

A231719 After zero, a(n) = largest m such that m! divides the difference between successive nodes A219666(n-1) and A219666(n) in the infinite trunk of the factorial beanstalk.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 2, 1, 1, 3, 2, 2, 1, 3, 1, 2, 1, 3, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 2, 1, 3, 1, 2, 1, 3, 1, 1, 1, 3, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 3, 1, 1, 1, 3, 2, 3, 2, 2, 1, 2, 1, 2, 1, 3, 1, 1, 1, 3, 2, 3, 2, 2, 1, 2, 3, 2, 1, 1, 1

Offset: 0

Antti Karttunen, Nov 12 2013

nonn

The first 4 occurs at n=2206. The first 5 occurs at n = 361788001015 = A226061(16).

Antti Karttunen, Table of n, a(n) for n = 0..21622

Cf. A055881, A219666, A226061, A230406, A230410, A231717, A232096.

nn = 1200; m = 1; While[Factorial@ m < nn, m++]; m; t = TakeWhile[
Reverse@ NestList[# - Total@ IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]] &, nn, 182], # <= 1000 &]; {0}~Join~Table[SelectFirst[Reverse@ Range@ 10, Divisible[t[[n]] - t[[n - 1]], #!] &], {n, 2, 87}] (* Michael De Vlieger, Jun 27 2016, Version 10.2 *)

(define (A231719 n) (if (zero? n) n (A055881 (A230406 n))))

a(0)=0 and for n>=1, a(n) = A055881(A230406(n)).

For all n, a(A226061(n+1)) = A232096(n).

A232741 Numbers n for which the largest m such that (m-1)! divides n is a prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76

Offset: 1

Antti Karttunen, Dec 01 2013

nonn

Numbers n for which A055881(n) is one of the terms of A006093.
Equally: Numbers n for which {the number of the trailing zeros in their factorial base representation A007623(n)} + 2 is a prime.
The sequence can be described in the following manner: Sequence includes all multiples of 1! and 2! (odd and even numbers), except that it excludes from those the multiples of 3! (6), except that it includes the multiples of 4! (24), except that it excludes the multiples of 5! (120), except that it includes the multiples of 6! (720), except that it excludes the multiples of 7! (5040) (and also those of 8! and 9!, because here 8+1 = 9 is the first odd composite), except that it includes the multiples of 10!, but excludes the multiples of 11!, but includes the multiples of 12!, but excludes the multiples of 13! (and also of 14! and 15!, because 14-16 are all composites), but includes the multiples of 16!, and so on, ad infinitum.

Antti Karttunen, Table of n, a(n) for n = 1..8748

Complement: A232742. Cf. A055926, A232744-A232745, A007623, A055881, A230403.

A232743 Numbers n for which the largest m such that (m-1)! divides n is a composite number > 5.

Original entry on oeis.org

120, 240, 360, 480, 600, 840, 960, 1080, 1200, 1320, 1560, 1680, 1800, 1920, 2040, 2280, 2400, 2520, 2640, 2760, 3000, 3120, 3240, 3360, 3480, 3720, 3840, 3960, 4080, 4200, 4440, 4560, 4680, 4800, 4920, 5040, 5160, 5280, 5400, 5520, 5640, 5880, 6000, 6120, 6240

Offset: 1

Antti Karttunen, Dec 01 2013

nonn

Numbers n for which A055881(n)>4 and is one of the terms of A072668.
Numbers n for which two plus the number of the trailing zeros in their factorial base representation A007623(n) is a composite number larger than 5.
All terms are multiples of 120. Specifically, these are all those terms of A232742 which are divisible by 120 (or equally: 24).
Please see also the comments in A055926, whose subset this sequence is.

			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.

Antti Karttunen, Table of n, a(n) for n = 1..25920

Subset of both A232742 and A055926.

Cf. also A007623, A055881, A230403, A232744-A232745.

cp's OEIS Frontend

A055874 a(n) = largest m such that 1, 2, ..., m divide n.

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A099563 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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A076934 Smallest integer of the form n/k!.

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A230403 a(n) = the largest k such that (k+1)! divides n; the number of trailing zeros in the factorial base representation of n (A007623(n)).

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A055926 Numbers k such that {largest m such that 1, 2, ..., m divide k} is different from {largest m such that m! divides k}; numbers k which are either odd multiples of 12 or the largest m such that (m-1)! divides k is a composite number > 5.

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Extensions

A232745 Numbers k for which the largest m such that m! divides k is even.

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Programs

Mathematica

A232744 Numbers k for which the largest m such that m! divides k is odd.

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