A067850 -id:A067850 - cp's OEIS frontend

A127032 Maximal value of m such that 5^m <= n! : a(n) = floor( log(n!) / log(5) ).

0, 0, 1, 1, 2, 4, 5, 6, 7, 9, 10, 12, 14, 15, 17, 19, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 55, 57, 59, 61, 63, 66, 68, 70, 73, 75, 77, 80, 82, 85, 87, 89, 92, 94, 97, 99, 102, 104, 107, 109, 112, 114, 117

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Cf. A067850, A127031, A127032, A127033, A127034, A127035, A127036, A127037, A127039.

With[{c=Log[5]},Table[Floor[Log[n!]/c],{n,60}]] (* Harvey P. Dale, Nov 16 2021 *)

A127033 Maximal value of m such that 7^m <= n!: a(n) = floor( log(n!) / log(7) ).

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 28, 29, 31, 33, 34, 36, 38, 40, 41, 43, 45, 47, 49, 51, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A067850, A127031, A127032, A127033, A127034, A127035, A127036, A127037, A127039.

[Floor(Log(Factorial(n)) / Log(7)) : n in [1..100]]; // Vincenzo Librandi, Dec 15 2016

Table[Floor[Log[n!]/Log[7]], {n,1,100}] (* G. C. Greubel, Dec 14 2016 *)

a(n)=lngamma(n+1)\log(7) \\ Charles R Greathouse IV, Sep 04 2015

a(n)=logint(n!,7) \\ Charles R Greathouse IV, Sep 04 2015

A127034 Maximal value of m such that 11^m <= n! : a(n) = floor( log(n!) / log(11) ).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 28, 29, 31, 32, 34, 35, 36, 38, 39, 41, 42, 44, 46, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 70, 71, 73, 75, 76, 78

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Cf. A067850, A127031, A127032, A127033, A127034, A127035, A127036, A127037, A127039.

a(n)=lngamma(n+1)\log(11) \\ Charles R Greathouse IV, Sep 04 2015

a(n)=logint(n!,11) \\ Charles R Greathouse IV, Sep 04 2015

A127035 Maximal value of m such that 13^m <= n! : a(n) = floor( log(n!) / log(13) ).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 40, 41, 43, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 59, 60, 62, 64, 65, 67, 68, 70, 71, 73

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Cf. A067850, A127031, A127032, A127033, A127034, A127035, A127036, A127037, A127039.

a(n)=lngamma(n+1)\log(13) \\ Charles R Greathouse IV, Sep 04 2015

a(n)=logint(n!,13) \\ Charles R Greathouse IV, Sep 04 2015

A127036 a(n) = maximal value of m such that 17^m divides n! (17^m <= n!).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 37, 38, 40, 41, 42, 44, 45, 46, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 62, 63, 65, 66

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Cf. A067850, A127031, A127032, A127033, A127034, A127035, A127036, A127037, A127039.

a(n)=lngamma(n+1)\log(17) \\ Charles R Greathouse IV, Sep 04 2015

a(n)=logint(n!,17) \\ Charles R Greathouse IV, Sep 04 2015

A127037 Maximal value of m such that 19^m <= n! : a(n) = floor( log(n!) / log(19) ).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Cf. A067850, A127031, A127032, A127033, A127034, A127035, A127036, A127037, A127039.

a(n)=lngamma(n+1)\log(19) \\ Charles R Greathouse IV, Sep 04 2015

a(n)=logint(n!,19) \\ Charles R Greathouse IV, Sep 04 2015

A127039 Maximal value of m such that 29^m <= n! : a(n) = floor( log(n!) / log(29) ).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 56

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Cf. A067850, A127031, A127032, A127033, A127034, A127035, A127036, A127037, A127039.

Table[Floor[Log[29,n!]],{n,60}] (* Harvey P. Dale, May 09 2016 *)

a(n)=lngamma(n+1)\log(29) \\ Charles R Greathouse IV, Sep 02 2015

a(n)=logint(n!,29) \\ Charles R Greathouse IV, Sep 04 2015

A084321 Least number k such that between k! and (k+1)! there are n powers of 2 (each interval includes (k+1)! but not k!).

Original entry on oeis.org

1, 3, 5, 10, 19, 35, 64, 139, 256, 536, 1061, 2095, 4169, 8282, 16517, 32903, 65646, 131205, 262579, 525083, 1048893, 2098826, 4195521, 8390583, 16782032, 33560609, 67118347, 134229613, 268453180, 536890474, 1073764782, 2147523518

Offset: 1

Labos Elemer, Jun 19 2003

nonn

a(n) is near the (n-1)th power of 2, the difference is A085355.

			a(3)=5 since between 5!=120 and 6!=720 is the first time 3 powers of 2 arise, namely, 128, 256 and 512.

Kevin Ryde, Table of n, a(n) for n = 1..750
Kevin Ryde, C Code

Cf. A067850, A058033, A000142, A000079, A084320, A084420, A085355.

C
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/* See links */
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LogBase2Stirling[n_] := N[ Log[2, 2 Pi n]/2 + n*Log[2, n/E] + Log[2, 1 + 1/(12n) + 1/(288n^2) - 139/(51840n^3) - 571/(2488320n^4) + 163879/(209018880n^5)], 64]; k = 1; Do[ While[ Floor[ LogBase2Stirling[k + 1]] - Floor[ LogBase2Stirling[k]] < n, k++ ]; Print[k], {n, 1, 33}]

a(n) = minimum x for which floor(log_2((x+1)!)) - floor(log_2(x!)) = n.

a(n) = minimum x for which A084320(x) = n.

Edited and extended by Robert G. Wilson v, Jun 24 2003
Definition clarified by Jianing Song, Aug 08 2022
a(26) corrected by Kevin Ryde, Apr 25 2024

A085301 Number of factorials between two primorials.

Original entry on oeis.org

2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1

Offset: 1

Labos Elemer, Jun 26 2003

nonn

Seems provable: a(n) > 0 for all n; seems more difficult to prove (if true at all) that a(n)=1 or 2; for n < 2050 it holds. Stirling's approximation and Prime Number Theorem together may help.

			n=1: between 1st (=2) and 2nd (=6) primorials, the factorials 2!=2 and 3!=6 occur, so a(1)=2.
n=2: between the primorials 6 and 30, the factorials 3!=6 and 4!=24 occur, so a(2)=2.
Factorial and primorial sets coincide only in case of n = 1,2: {2,6}.
If n > 3, factorials are never squarefree; but primorials are always squarefree, so they are disjoint.
n=5: between the 5th and 6th primorials 2310 and 30030, only the factorial 7!=5040 occurs.
n=6: between the primorials 30030 and 510510, the factorials 8!=40320 and 9!=362880 occur.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000142, A002110, A067850, A084320, A084321, A084558, A084972, A085355.

fn[n_] := Module[{k = 1, r = n}, While[r >= 1, k++; r /= k]; k - 1];
prim[n_] := Times @@ Prime[Range[n]];
a[n_] := fn[prim[n]] - fn[prim[n - 1]]; a[1] = a[2] = 2; Array[a, 100] (* Amiram Eldar, Oct 24 2024 *)

a(n) = Card[{k; q(n) <= k! <= q(n+1)}, where q(j) = A002110(j), the j-th primorial; closed intervals required only for n = 1, 2.

a(n) = A084558(A002110(n)) - A084558(A002110(n-1)) for n >= 3. - Amiram Eldar, Oct 24 2024

A084320 Number of powers of two between 2 consecutive factorials (2! including).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 5, 5, 4, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5, 5, 6, 5, 5, 6, 5, 6, 5, 6, 5, 6, 6, 5, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 5, 6, 7, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 7, 6, 6, 7, 6, 6, 7, 6, 7, 6, 7, 6, 6, 7, 7, 6, 7, 6, 7, 6, 7, 7, 6, 7, 7, 6, 7, 7, 6

Offset: 1

Labos Elemer, Jun 19 2003

nonn

			n=7: a(7)=3 because between 5040 and 40320 three powers of 2 occur: 8192, 16384 and 32768.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A067850, A058033, A000142, A000079.

Table[Floor[Log[2, (w+1)! ]//N]-Floor[Log[2, w! ]//N], {w, 1, 128}]

a(n)=if(n<6,(n+1)\2,log((n+1)!)\log(2)-log(n!)\log(2)) \\ Charles R Greathouse IV, Dec 26 2013

a(n) = A067850(n+1) - A067850(n).

a(n) = A000523(n) + O(1). - Charles R Greathouse IV, Dec 26 2013

cp's OEIS Frontend

A127032 Maximal value of m such that 5^m <= n! : a(n) = floor( log(n!) / log(5) ).

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A127033 Maximal value of m such that 7^m <= n!: a(n) = floor( log(n!) / log(7) ).

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Magma

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A127034 Maximal value of m such that 11^m <= n! : a(n) = floor( log(n!) / log(11) ).

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PARI

A127035 Maximal value of m such that 13^m <= n! : a(n) = floor( log(n!) / log(13) ).

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PARI

A127036 a(n) = maximal value of m such that 17^m divides n! (17^m <= n!).

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PARI

PARI

A127037 Maximal value of m such that 19^m <= n! : a(n) = floor( log(n!) / log(19) ).

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PARI

PARI

A127039 Maximal value of m such that 29^m <= n! : a(n) = floor( log(n!) / log(29) ).

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Mathematica

PARI

PARI

A084321 Least number k such that between k! and (k+1)! there are n powers of 2 (each interval includes (k+1)! but not k!).

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C

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A085301 Number of factorials between two primorials.

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