A074181 -id:A074181 - cp's OEIS frontend

A060151 Number of base n digits required to write n!.

1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 19, 19, 20, 21, 22, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 34, 35, 36, 37, 38, 38, 39, 40, 41, 42, 43, 43, 44, 45, 46, 47, 47, 48, 49, 50, 51, 51, 52, 53, 54, 55, 56, 56, 57

Offset: 1

Henry Bottomley, Mar 08 2001

			a(6)=4 since 6!=720, which in base 6 is 3200.

Harry J. Smith and Danny Rorabaugh, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. A011776 for number of final zeros of n! written in base n.

Cf. A039960, A074181-A074184.

[1] cat [1 + Floor(Log(Factorial(n))/Log(n)): n in [2..80]]; // Vincenzo Librandi, Apr 15 2015

Join[{1},Table[IntegerLength[n!,n],{n,2,80}]] (* Harvey P. Dale, May 30 2014 *)

a(n)=if(n>1, logint(n!,n), 1) \\ Charles R Greathouse IV, Oct 29 2016

a(n)=if(n>1, lngamma(n+1)\log(n))+1 \\ Charles R Greathouse IV, Oct 29 2016

[1] + [1 + floor(log(factorial(n))/log(n)) for n in range(2,74)] # Danny Rorabaugh, Apr 14 2015

a(n) = 1 + floor(log(n!)/log(n)) = 1 + A039960(n) for n>1.
From Danny Rorabaugh, Apr 14 2015: (Start)
a(n) = 1 + log_n(A074182(n)) for n>1.
a(n) = A074184(n) = log_n(A074181(n)) for n>2.
(End)
a(n) = n - n/log n + O(1). - Charles R Greathouse IV, Oct 29 2016

A074184 Index of the smallest power of n >= n!.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 19, 19, 20, 21, 22, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 34, 35, 36, 37, 38, 38, 39, 40, 41, 42, 43, 43, 44, 45, 46, 47, 47, 48, 49, 50, 51, 51, 52, 53, 54, 55, 56, 56, 57

Offset: 1

Amarnath Murthy, Aug 31 2002

Essentially the same as A060151. - R. J. Mathar, Dec 15 2008

Danny Rorabaugh, Table of n, a(n) for n = 1..10000

Cf. A039960, A074181, A074182.

[1] cat [Ceiling(Log(Factorial(n))/Log(n)): n in [2..80]]; // Vincenzo Librandi, Apr 15 2015

a(n)=if(n>2,lngamma(n+1)\log(n))+1 \\ Charles R Greathouse IV, Sep 02 2015

[1]+[ceil(log(factorial(n))/log(n)) for n in range(2, 74)] # Danny Rorabaugh, Apr 14 2015

From Danny Rorabaugh, Apr 14 2015: (Start)
a(n) = log_n(A074181(n)).
a(n) = ceiling(log_n(n!)) for n>1.
a(n) = A060151(n) = 1 + A039960(n) = 1 + log_n(A074182(n)) for n>2.
(End)

More terms from Jason Earls, Sep 02 2002

A039960 For n >= 2, a(n) = largest value of k such that n^k is <= n! (a(0) = a(1) = 1 by convention).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 14, 14, 15, 16, 17, 18, 18, 19, 20, 21, 21, 22, 23, 24, 25, 25, 26, 27, 28, 29, 29, 30, 31, 32, 33, 33, 34, 35, 36, 37, 37, 38, 39, 40, 41, 42, 42, 43, 44, 45, 46, 46, 47, 48, 49, 50, 50, 51, 52, 53, 54, 55, 55, 56, 57

Offset: 0

Dan Bentley (bentini(AT)yahoo.com)

Seems to be slightly more than (but asymptotic to) number of nonprimes less than or equal to n.

			a(7)=4 because 7! = 5040, 7^4 = 2401 but 7^5 = 16807.
a(6)=3 since 6^3.67195... = 720 = 6! and 6^3 <= 6! < 6^4, i.e., 216 <= 720 < 1296.

Danny Rorabaugh, Table of n, a(n) for n = 0..10000

Cf. A011776, A074181, A074182, A074184.

[1,1] cat [Floor(Log(Factorial(n))/Log(n)): n in [2..80]]; // Vincenzo Librandi, Apr 15 2015

ds[x_, y_] :=y!-y^x; a[n_] :=Block[{m=1, s=ds[m, n]}, While[Sign[s]!=-1&&!Greater[m, 256], m++ ];m]; Table[a[n]-1, {n, 3, 200}]
(* or *)
Table[Count[Part[Sign[Table[Table[n!-n^j, {j, 1, 128}], {n, 1, 128}]], u], 1], {u, 1, 128}] (* Labos Elemer *)
Join[{1,1},Table[Floor[Log[n,n!]],{n,2,80}]] (* Harvey P. Dale, Sep 24 2019 *)

a(n)=if(n>3,lngamma(n+1)\log(n),1) \\ Charles R Greathouse IV, Sep 02 2015

[1,1] + [floor(log(factorial(n))/log(n)) for n in range(2,75)] # Danny Rorabaugh, Apr 14 2015

a(n) = floor(log_n(n!)) for n > 1.
a(n) = A060151(n) - 1 for n > 1. - Henry Bottomley, Mar 08 2001
From Danny Rorabaugh, Apr 14 2015: (Start)
a(n) = log_n(A074182(n)) for n > 1.
a(n) = A074184 - 1 = log_n(A074181(n)) - 1 for n > 2. (End)
From Robert Israel, Apr 14 2015: (Start)
n*(1-1/log(n)) + 1 > log(n!)/log(n) > n*(1-1/log(n)) for n >= 7.
Thus a(n) is either floor(n*(1-1/log(n))) or ceiling(n*(1-1/log(n))) for n >= 7 (and in fact this is the case for n >= 3). (End)

Corrected and extended by Henry Bottomley, Mar 08 2001

Edited by N. J. A. Sloane, Sep 26 2008 at the suggestion of R. J. Mathar

A074182 Largest power of n <= n!.

Original entry on oeis.org

1, 2, 3, 16, 25, 216, 2401, 32768, 59049, 1000000, 19487171, 429981696, 815730721, 20661046784, 576650390625, 17592186044416, 34271896307633, 1156831381426176, 42052983462257059, 1638400000000000000

Offset: 1

Amarnath Murthy, Aug 31 2002

Alois P. Heinz, Table of n, a(n) for n = 1..400

Cf. A039960, A060151, A074181.

[1] cat [n^Floor(Log(Factorial(n)) / Log(n)): n in [2..25]]; // Vincenzo Librandi, Apr 15 2015

a(n)=if(n>3,n^logint(n!,n),n) \\ Charles R Greathouse IV, Oct 11 2015

[1] + [n^(floor(log(factorial(n))/log(n))) for n in range(2, 21)] # Danny Rorabaugh, Apr 14 2015

From Danny Rorabaugh, Apr 14 2015: (Start)
a(n) = n^A039960(n) = n^(A060151(n)-1).
a(n) = n^floor(log_n(n!)) for n>1.
a(n) = A074181(n)/n for n>2.
(End)

More terms from Jason Earls, Sep 02 2002

A248868 Exponents n that make k! < k^n < (k+1)! hold true for some integer k > 1, in increasing order by k, then n (if applicable).

Original entry on oeis.org

2, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 19, 19, 20, 21, 22, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 34, 35, 36, 37, 38, 38, 39, 40, 41, 42, 43, 43, 44, 45, 46, 47, 47, 48, 49, 50, 51, 51, 52, 53, 54

Offset: 1

Juan Castaneda, Mar 04 2015

nonn

This sequence consists of those positive integers that, when taken as exponents of some positive integer greater than 1, make the corresponding power of that other integer fall strictly between its factorial and the factorial of the next integer, as shown in the examples.
The sequence { floor(log_n((n+1)!)) | n>=2 } is a subsequence.
This sequence is nondecreasing. Indeed for k>1, k^n<(k+1)! implies n<=k, which implies ((k+1)/k)^(n-1) <= (1 + 1/k)^(k-1) = Sum_{i=0..k-1} binomial(k-1,i) (1/k)^i < Sum_{i=0..k-1} ((k-1)/k)^i < k, which implies (k+1)^(n-1)Danny Rorabaugh, Apr 03 2015
From Danny Rorabaugh, Apr 15 2015: (Start)
This sequence is the same as A074184 for 6<=n<=10000.
For k > 2, k! < k^(ceiling(log_k(k!))) < (k+1)!.
The two sequences continue to be identical provided k^(1 + ceiling(log_k(k!))) > (k+1)! when k > 5.
This is equivalent to k^(2 - fractional_part(log_k(k!))) > k + 1, which can be approximated by fractional_part(1/2 - (k + sqrt(2*Pi))/log(k)) < 1 - 1/(k*log(k)) using Stirling's approximation.
Are either of the final inequalities true for all sufficiently large k?
(End)

			2! < 2^2 < 3! < 3^2 < 4! < 4^3 < 5! < 5^3 < 5^4 < 6! < 6^4 < 7! < 7^5 < 8! and so on; this sequence consists of the exponents.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000

Cf. A060151, A074181-A074184, A111683.

[x for sublist in [[k for k in [0..ceil(log(factorial(n+1),base=n))] if (factorial(n)Tom Edgar, Mar 04 2015

More terms from Tom Edgar, Mar 04 2015

A256774 All factorials n! along with powers of the numbers n and n+1 that fall in between n! and (n+1)!, in increasing order.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 16, 24, 25, 64, 120, 125, 216, 625, 720, 1296, 2401, 5040, 16807, 32768, 40320, 59049, 262144, 362880, 531441, 1000000, 3628800, 10000000, 19487171, 39916800, 214358881, 429981696, 479001600, 815730721, 5159780352, 6227020800, 10604499373, 20661046784, 87178291200, 289254654976

Offset: 1

Juan Castaneda, Apr 10 2015

For each positive integer n, we consider the two factorials n! and (n+1)! as lower and upper bounds of an interval. Then we look for all powers of n and all powers of n+1 that fall inside that interval. We sort those numbers in increasing order, and we append them to the sequence without allowing duplicates. Then we move on to the next integer, and so on.

A000142 (without its first term that stands for 0!) is a subsequence.

			With n=1: 1! < 2! gives a(1)=1, a(2)=2.
With n=2: 2! < 3^1 < 2^2 < 3! gives a(3)=3, a(4)=4, a(5)=6.
With n=3: 3! < 3^2 < 4^2 < 4! gives a(6)=9, a(7)=16, a(8)=24.
With n=4: 4! < 5^2 < 4^3 < 5! gives a(9)=25, a(10)=64, a(11)=120.
With n=5: 5! < 5^3 < 6^3 < 5^4 < 6! gives a(12)=125, a(13)=216, a(14)=625, a(15)=720

Michael De Vlieger, Table of n, a(n) for n = 1..1346

Cf. A000142, A039960, A060151, A074181, A074182, A074184, A111683.

f[n_] := Block[{a = n!, b = (n + 1)!}, Sort@ Union[{a}, n^Range[Ceiling@ Log[n, a], Floor@ Log[n, b]], (n + 1)^Range[Ceiling@ Log[n + 1, a], Floor@ Log[n + 1, b]]]]; {1}~Join~(f /@ Range[2, 14] // Flatten) (* Michael De Vlieger, Apr 15 2015 *)

tabf(nn) = {print([1]); for (n=2, nn, v = [n!]; ka = ceil(log(n!+1)/log(n)); kb = floor(log((n+1)!-1)/log(n)); for (k=ka, kb, v = concat(v, n^k);); ka = ceil(log(n!+1)/log(n+1)); kb = floor(log((n+1)!-1)/log(n+1)); for (k=ka, kb, v = concat(v, (n+1)^k);); print(vecsort(v));); } \\ Michel Marcus, Apr 22 2015

cp's OEIS Frontend

A060151 Number of base n digits required to write n!.

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Magma

Mathematica

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PARI

Sage

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A074184 Index of the smallest power of n >= n!.

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Magma

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A039960 For n >= 2, a(n) = largest value of k such that n^k is <= n! (a(0) = a(1) = 1 by convention).

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Magma

Mathematica

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Sage

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A074182 Largest power of n <= n!.

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Magma

PARI

Sage

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Extensions

A248868 Exponents n that make k! < k^n < (k+1)! hold true for some integer k > 1, in increasing order by k, then n (if applicable).

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Sage

Extensions

A256774 All factorials n! along with powers of the numbers n and n+1 that fall in between n! and (n+1)!, in increasing order.

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Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI