A232743 -id:A232743 - cp's OEIS frontend

A055881 a(n) = largest m such that m! divides n.

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

Offset: 1

Leroy Quet and Labos Elemer, Jul 16 2000

Number of factorial divisors of n. - Amarnath Murthy, Oct 19 2002
The sequence may be constructed as follows. Step 1: start with 1, concatenate and add +1 to last term gives: 1,2. Step 2: 2 is the last term so concatenate twice those terms and add +1 to last term gives: 1, 2, 1, 2, 1, 3 we get 6 terms. Step 3: 3 is the last term, concatenate 3 times those 6 terms and add +1 to last term gives: 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, iterates. At k-th step we obtain (k+1)! terms. - Benoit Cloitre, Mar 11 2003
From Benoit Cloitre, Aug 17 2007, edited by M. F. Hasler, Jun 28 2016: (Start)
Another way to construct the sequence: start from an infinite series of 1's:
   1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... Replace every second 1 by a 2 giving:
   1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ... Replace every third 2 by a 3 giving:
   1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, ... Replace every fourth 3 by a 4 etc. (End)
This sequence is the fixed point, starting with 1, of the morphism m, where m(1) = 1, 2, and for k > 1, m(k) is the concatenation of m(k - 1), the sequence up to the first k, and k + 1. Thus m(2) = 1, 2, 1, 3; m(3) = 1, 2, 1, 3, 1, 2, 1, 2, 1, 4; m(4) = 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 5, etc. - Franklin T. Adams-Watters, Jun 10 2009
All permutations of n elements can be listed as follows: Start with the (arbitrary) permutation P(0), and to obtain P(n + 1), reverse the first a(n) + 1 elements in P(n). The last permutation is the reversal of the first, so the path is a cycle in the underlying graph. See example and fxtbook link. - Joerg Arndt, Jul 16 2011
Positions of rightmost change with incrementing rising factorial numbers, see example. - Joerg Arndt, Dec 15 2012
Records appear at factorials. - Robert G. Wilson v, Dec 21 2012
One more than the number of trailing zeros (A230403(n)) in the factorial base representation of n (A007623(n)). - Antti Karttunen, Nov 18 2013
A062356(n) and a(n) coincide quite often. - R. J. Cano, Aug 04 2014
For n>0 and 1<=j<=(n+1)!-1, (n+1)^2-1=A005563(n) is the number of times that a(j)=n-1. - R. J. Cano, Dec 23 2016

			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)

Antti Karttunen, Table of n, a(n) for n = 1..10080
Joerg Arndt, Matters Computational (The Fxtbook), section 10.4, pp.245-248 (prefix reversals); section 10.5, pp. 248-250 (Heap's method).
R. J. Cano, Alternative sequencer (PARI/GP).
Claude Lenormand, Comments on this sequence.
József Sándor, On Additive Analogues of Certain Arithmetic Smarandache Functions.
Index entries for sequences related to factorial base representation.

Cf. A055874, A055926, A055770, A062356, A073575, A091131, A230403, A230404, A230405, A076733, A232096, A232098, A233285, A233267, A233269, A231719, A232741, A232742, A232743, A232744, A232745, A060832 (partial sums).
This sequence occurs also in the next to middle diagonals of A230415 and as the second rightmost column of triangle A230417.
Other sequences related to factorial base representation (A007623): A034968, A084558, A099563, A060130, A227130, A227132, A227148, A227149, A153880.
Analogous sequence for binary (base-2) representation: A001511.

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 *)

PARI
```
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))))))

G.f.: Sum_{k > 0} x^(k!)/(1 - x^(k!)). - Vladeta Jovovic, Dec 13 2002
a(n) = A230403(n)+1. - Antti Karttunen, Nov 18 2013
a(n) = A230415(n-1,n) = A230415(n,n-1) = A230417(n,n-1). - Antti Karttunen, Nov 19 2013
a(m!+n) = a(n) if 1 <= n <= m*m! - 1 = A001563(m) - 1. - R. J. Cano, Jun 27 2016
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = e - 1 (A091131). - Amiram Eldar, Jul 23 2022

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))).

A232099 Numbers n such that {largest m such that 1, 2, ..., m divide n} is different from {largest m such that m! divides n^2}.

Original entry on oeis.org

840, 2520, 4200, 5880, 7560, 9240, 10920, 12600, 14280, 15960, 17640, 19320, 21000, 22680, 24360, 26040, 27720, 29400, 31080, 32760, 34440, 36120, 37800, 39480, 41160, 42840, 44520, 46200, 47880, 49560, 51240, 52920, 54600, 55440, 56280, 57960, 59640, 61320, 63000

Offset: 1

Antti Karttunen, Nov 18 2013

nonn

Numbers n such that A055874(n) differs from A232098(n). (By the definition of the sequence).
This sequence is a subset of A055926. Please see there for a proof. From that follows that A055881(a(n))+1 is always composite (in range n=1..100000, only values 6, 8, 9 and 10 occur).
Also, incidentally, for the first five terms, n=1..5, a(n) = 70*A055926(n), then a(6)=77*A055926(6), and the next time the ratio A232099(n)/A055926(n) is integral is at n=21, where a(n) = 82*A055926(21), at n=41  (a(41) = 79*A055926(41) = 79*840 = 66360), at n=136, a(136) = 80*A055926(136) = 80*2772 = 221760 and at n=1489, where a(1489) = 80*A055926(1489) = 80 * 30492 = 2439360. The ratio seems to converge towards some value a little less than 80. Please see the plot generated by Plot2 in the links section.

			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.

Antti Karttunen, Table of n, a(n) for n = 1..10000
OEIS Server, Ratio A232099(n)/A055926(n) plotted with Plot 2
OEIS Server, Ratio A232099(n)/A232743(n) plotted with Plot 2
Wikipedia, Wilson's theorem (Please see especially the section "Composite modulus")

Subset of A055926.

Cf. A055881, A055874, A232098, A232100.

For all n, a(n) = A055926(A232100(n)). [Follows from the definition of A232100, but cannot as such be used to compute the sequence. Use the given Scheme-program instead.]

A232742 Numbers n for which the largest m such that (m-1)! divides n is a composite.

Original entry on oeis.org

6, 12, 18, 30, 36, 42, 54, 60, 66, 78, 84, 90, 102, 108, 114, 120, 126, 132, 138, 150, 156, 162, 174, 180, 186, 198, 204, 210, 222, 228, 234, 240, 246, 252, 258, 270, 276, 282, 294, 300, 306, 318, 324, 330, 342, 348, 354, 360, 366, 372, 378, 390, 396, 402, 414

Offset: 1

Antti Karttunen, Dec 01 2013

nonn

Numbers n for which A055881(n) is one of the terms of A072668.
Equally: numbers n for which {the number of the trailing zeros in their factorial base representation A007623(n)} + 2 is a composite number.
All terms are divisible by 6.
The sequence can be described in the following manner: Sequence includes all multiples of 3!, except that it excludes from those 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) (and also those of 8! and 9!, because here 8+1 = 9 is the first odd composite), of which it however excludes the multiples of 10!, except that it includes the multiples of 11!, but excludes the multiples of 12!, but includes the multiples of 13! (and 14! and 15!, because 14-16 are all composites), but excludes the multiples of 16!, and so on, ad infinitum.

			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.

Antti Karttunen, Table of n, a(n) for n = 1..10656

Complement: A232741. Subset: A232743.

Cf. A055926, A232744-A232745, A007623, A055881, A230403.

cp's OEIS Frontend

A055881 a(n) = largest m such that m! divides 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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A232745 Numbers k for which the largest m such that m! divides k is even.

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

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A232099 Numbers n such that {largest m such that 1, 2, ..., m divide n} is different from {largest m such that m! divides n^2}.

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A232742 Numbers n for which the largest m such that (m-1)! divides n is a composite.

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