A055881 -id:A055881 - cp's OEIS frontend

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}.

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.]

A060832 a(n) = Sum_{k>0} floor(n/k!).

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 10, 11, 13, 14, 16, 17, 20, 21, 23, 24, 26, 27, 30, 31, 33, 34, 36, 37, 41, 42, 44, 45, 47, 48, 51, 52, 54, 55, 57, 58, 61, 62, 64, 65, 67, 68, 71, 72, 74, 75, 77, 78, 82, 83, 85, 86, 88, 89, 92, 93, 95, 96, 98, 99, 102, 103, 105, 106, 108, 109, 112, 113

Offset: 0

Henry Bottomley, May 01 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A013936, A013939, A038663, A055881.

[0] cat [&+[Floor(m/Factorial(k)):k in [1..m]]:m in [1..70]]; // Marius A. Burtea, Jul 11 2019

a(n)={my(s=0, d=1, f=1); while (n>=d, s+=n\d; f++; d*=f); s} \\ Harry J. Smith, Jul 12 2009

a(n) = round(sumpos(k=1, n\k!)); \\ Michel Marcus, Jan 24 2025

a(n) = a(n-1) + A055881(n).
a(n) = (e-1)*n + f(n) where f(n) < 0. - Benoit Cloitre, Jun 19 2002
f is unbounded in the negative direction. The assertion that f(n) < 0 is correct, since (e-1)*n = Sum_{k>=1} n/k! is term for term >= this sequence. - Franklin T. Adams-Watters, Nov 03 2005
G.f.: (1/(1 - x)) * Sum_{k>=1} x^(k!)/(1 - x^(k!)). - Ilya Gutkovskiy, Jul 11 2019

A207324 List of permutations of 1,2,3,...,n for n=1,2,3,..., in the order they are output by Steinhaus-Johnson-Trotter algorithm.

Original entry on oeis.org

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

Offset: 1

R. J. Cano, Sep 14 2012

This table is otherwise similar to A030298, but lists permutations in the order given by the Steinhaus-Trotter-Johnson algorithm. - Antti Karttunen, Dec 28 2012

			For the set of the first two natural numbers {1,2} the unique permutations possible are 12 and 21, concatenated with 1 for {1} the resulting sequence would be 1, 1, 2, 2, 1.
If we consider up to 3 elements {1,2,3}, we have 123, 132, 312, 321, 231, 213 and the concatenation gives: 1, 1, 2, 2, 1, 1, 2, 3, 1, 3, 2, 3, 1, 2, 3, 2, 1, 2, 3, 1, 2, 1, 3.
Up to N concatenations, the sequence will have a total of Sum_{k=1..N} (k! * k) = (N+1)! - 1 = A033312(N+1) terms.

R. J. Cano, Table of n, a(n) for n = 1..10000
Joerg Arndt, C programs related to this sequence
R. J. Cano, Sequencer programs and additional information
Selmer M. Johnson, Generation of permutations by adjacent transposition, Mathematics of Computation, 17 (1963), p. 282-285.
Wikipedia, Steinhaus-Johnson-Trotter algorithm
Index entries for sequences related to permutations

Cf. A030298, A055881.
Cf. A001563 (row lengths), A001286 (row sums).
Pair (A130664(n),A084555(n)) = (1,1),(2,3),(4,5),(6,8),(9,11),(12,14),... gives the starting and ending offsets of the n-th permutation in this list.

Edited by N. J. A. Sloane, Antti Karttunen and R. J. Cano

A230417 Lower triangular region of A230415, a triangular table read by rows: T(n, k) tells in how many digit positions the factorial base representations (A007623) of n and k differ, where (n, k) = (0,0), (1,0), (1,1), (2,0), (2,1), (2,2), ..., n >= 0 and (0 <= k <= n).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Nov 10 2013

			This triangular table begins:
  0;
  1, 0;
  1, 2, 0;
  2, 1, 1, 0;
  1, 2, 1, 2, 0;
  2, 1, 2, 1, 1, 0;
  1, 2, 2, 3, 2, 3, 0;
  ...
Please see A230415 for examples showing how the terms are computed.

Antti Karttunen, Rows n = 0..120 of triangle, flattened

This is a lower, or equivalently, an upper triangular subregion of symmetric square array A230415.

Cf. A231714, A060130, A055881.

(define (A230417 n) (A230415bi (A003056 n) (A002262 n)))
(define (A230415bi x y) (let loop ((x x) (y y) (i 2) (d 0)) (cond ((and (zero? x) (zero? y)) d) (else (loop (floor->exact (/ x i)) (floor->exact (/ y i)) (+ i 1) (+ d (if (= (modulo x i) (modulo y i)) 0 1)))))))

a(n) = A230415bi(A003056(n),A002262(n)). [As a sequence, this is obtained by taking a subsection from array A230415.]
T(n,0) = A060130(n) [the leftmost column].
For n >= 1, T(n,n-1) = A055881(n) [the last nonzero column].
Each entry T(n,k) <= A231714(n,k).

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.

A230404 a(n) = the largest k such that (k+1)! divides 2n; the number of trailing zeros in the factorial base representation of even numbers.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 31 2013

Antti Karttunen, Table of n, a(n) for n = 1..10080

Used to compute A230405 and A219650. See A007623 for factorial base representation.
Analogous sequence for binary system: A001511.
Cf. A019762.

Scheme
```
(define (A230404 n) (A230403 (* 2 n)))
```

a(n) = A230403(2n) = A055881(2n) - 1.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*e - 4 = A019762 - 4 = 1.436563... . - Amiram Eldar, Jan 05 2024

A055770 Largest factorial number which divides n.

Original entry on oeis.org

1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 24, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 24, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 24, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 24, 1, 2, 1, 2, 1, 6

Offset: 1

Labos Elemer, Jul 12 2000

Largest m! which divides n.

			3! = 6 divides 12, so a(12) = 6.

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

Cf. A000142, A055881 (values of the m's), A055926, A055874, A073575.

Cf. also A053589.

With[{rf=Reverse[Range[7]!]},Table[SelectFirst[rf,Divisible[n,#]&],{n,120}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 05 2017 *)

A055770(n) = { my(m=1, i=2); while(!(n%m), m *= i; i++); return(m/(i-1)); } \\ Antti Karttunen, Dec 19 2018

a(n) = A000142(A055881(n)). - Antti Karttunen, Dec 19 2018

Name changed, old name moved to comments by Antti Karttunen, Dec 19 2018

A073575 Sum of factorial numbers dividing n.

Original entry on oeis.org

1, 3, 1, 3, 1, 9, 1, 3, 1, 3, 1, 9, 1, 3, 1, 3, 1, 9, 1, 3, 1, 3, 1, 33, 1, 3, 1, 3, 1, 9, 1, 3, 1, 3, 1, 9, 1, 3, 1, 3, 1, 9, 1, 3, 1, 3, 1, 33, 1, 3, 1, 3, 1, 9, 1, 3, 1, 3, 1, 9, 1, 3, 1, 3, 1, 9, 1, 3, 1, 3, 1, 33, 1, 3, 1, 3, 1, 9, 1, 3, 1, 3, 1, 9, 1, 3, 1, 3, 1, 9, 1, 3, 1, 3, 1, 33, 1, 3, 1, 3, 1, 9

Offset: 1

Vladeta Jovovic, Aug 27 2002

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10079

Cf. A000142, A007489, A055770, A055881, A007489.

Table[Total[Select[Table[i!, {i, n}], Divisible[n, #] &]], {n, 102}] (* Jayanta Basu, Jul 01 2013 *)

A073575(n) = { my(m=1, i=2, s=0); while(!(n%m), s += m; m *= i; i++); return(s); } \\ Antti Karttunen, Dec 19 2018

G.f.: Sum_{k>0} k!*x^(k!)/(1-x^(k!)). - Vladeta Jovovic, Dec 13 2002

a(n) = A007489(A055881(n)). - Antti Karttunen, Dec 19 2018

A076733 Largest k such that k! divides C(2n,n).

Original entry on oeis.org

2, 3, 2, 2, 3, 3, 4, 3, 2, 2, 4, 2, 2, 5, 6, 3, 3, 3, 5, 3, 5, 5, 6, 3, 4, 4, 2, 2, 5, 2, 2, 3, 3, 3, 4, 2, 2, 5, 2, 2, 5, 5, 7, 5, 7, 7, 7, 3, 5, 4, 4, 4, 7, 5, 4, 4, 4, 5, 6, 4, 4, 4, 5, 3, 3, 3, 5, 3, 5, 5, 6, 3, 5, 5, 7, 5, 7, 7, 7, 3, 2, 2, 5, 2, 2, 7, 5, 5, 7, 2, 2, 5, 2, 2, 7, 3, 5, 5, 5, 5, 6, 5, 5, 5, 6

Offset: 1

Benoit Cloitre, Oct 28 2002

nonn

All a(n) >= 2, with a(n) = 2 if and only if n is in A005836. - Robert Israel, Feb 01 2019

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005836, A055881.

f:= proc(n) local x,k;
  x:= binomial(2*n,n);
  for k from 2 do if not (x/k!)::integer then return k-1 fi od
end proc:
map(f, [$1..105]); # Robert Israel, Feb 01 2019

a[n_] := Module[{k = 2}, While[Divisible[Binomial[2n, n], k!], k++]; k-1];
Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Oct 01 2024 *)

a(n)=if(n<0,0,k=1; while(binomial(2*n,n)%(k!) == 0,k++); k-1)

A331171 a(n) = min(n, A225901(n)), where A225901 is factorial base flip.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 14, 15, 6, 7, 10, 11, 8, 9, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 48, 49, 52, 53, 50, 51, 66, 67, 70, 71, 68, 69, 60, 61, 64, 65, 62, 63, 54, 55, 58, 59

Offset: 0

Antti Karttunen, Jan 12 2020

For all i, j:
  a(i) = a(j) => A060130(i) = A060130(j).
For all i, j > 0:
  a(i) = a(j) => A055881(i) = A055881(j).

Antti Karttunen, Table of n, a(n) for n = 0..40319
Index entries for sequences related to factorial base representation

Cf. A055881, A060130, A225901.

Cf. also A330740, A331172.

A225901(n) = { my(s=0, d, k=2); while(n, d=n%k; n=n\k; if(d, s += (k-d)*(k-1)!); k=k+1); (s); };
A331171(n) = min(n, A225901(n));

a(n) = min(n, A225901(n)).

cp's OEIS Frontend

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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A060832 a(n) = Sum_{k>0} floor(n/k!).

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A207324 List of permutations of 1,2,3,...,n for n=1,2,3,..., in the order they are output by Steinhaus-Johnson-Trotter algorithm.

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A230417 Lower triangular region of A230415, a triangular table read by rows: T(n, k) tells in how many digit positions the factorial base representations (A007623) of n and k differ, where (n, k) = (0,0), (1,0), (1,1), (2,0), (2,1), (2,2), ..., n >= 0 and (0 <= k <= n).

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Scheme

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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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A230404 a(n) = the largest k such that (k+1)! divides 2n; the number of trailing zeros in the factorial base representation of even numbers.

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Scheme

Formula

A055770 Largest factorial number which divides n.

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Mathematica

PARI

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Extensions

A073575 Sum of factorial numbers dividing n.

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Mathematica

PARI

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