A011776 -id:A011776 - cp's OEIS frontend

A011778 Numbers k where A011776(k) grows.

2, 6, 12, 24, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360, 480, 630, 672, 720, 1080, 1120, 1260, 1344, 1440, 1890, 2016, 2160, 2240, 2520, 2880, 3024, 3360, 3780, 4032, 4320, 5040, 6048

Offset: 1

Robert G. Wilson v

Equivalently, 2 along with numbers A092427(j) where A092427(j) > A092427(j-1), for j > 0. - Derek Orr, Apr 16 2015
Is 45 the only odd number in this sequence? - Derek Orr, Apr 16 2015
a(184) = 212837625 and a(311) = 638512875 are the only other odd terms < 10^11. - Charlie Neder, Jan 27 2019

Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, Wiley NY 1991.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 251.

Charlie Neder, Table of n, a(n) for n = 1..697 (terms < 10^11; first 100 terms from T. D. Noe)
Index entries for sequences related to factorial numbers

Cf. A011776, A011777, A133481, A092427.

a(n) = A061769(n) + 1.

a(n) = A092427(A061770(n)). - Derek Orr, Apr 16 2015

Corrected by Vladeta Jovovic, Dec 01 2002

A373725 Numbers k such that A011776(k) = A011776(k+1).

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 15, 27, 63, 195, 728, 1443, 3843, 5475, 6174, 11913, 13376, 24963, 37635, 77283, 98595, 113398, 158403, 178083, 209763, 293763, 294335, 319124, 376995, 406503, 438243, 454275, 538755, 574563, 770883, 996003, 1196835, 1331715, 1444803, 1473795

Offset: 1

Amiram Eldar, Jun 15 2024

nonn

The corresponding values of A011776 are 1, 1, 1, 1, 2, 2, 3, 4, 10, 16, 60, ... .
All the terms above 3 are composite numbers since A011776(k) = 1 if and only if k = 4 or a prime.
Are there 3 consecutive integers above 8 that have an equal value of A011776? There are none below 10^10.
Conjecture: if p != 3 is a prime such that 2*p-1 is also a prime (p is in A005382 \ {3}), then 4*p^2 - 1 is a term of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A005382, A011776.

s[1] = 1; s[n_] := IntegerExponent[n!, n]; seq[kmax_] := Module[{v = {}, s1 = s[1]}, Do[s2 = s[k]; If[s1 == s2, AppendTo[v, k-1]]; s1 = s2, {k, 2, kmax}]; v]; seq[10^4]

lista(kmax) = {my(s1 = 1, s2); for(k = 2, kmax, s2 = valuation(k!, k); if(s1 == s2, print1(k-1, ", ")); s1 = s2);}

A090622 Square array read by antidiagonals of highest power of k dividing n! (with n,k>1).

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 0, 1, 3, 0, 0, 1, 1, 4, 0, 1, 0, 1, 2, 4, 0, 0, 1, 1, 2, 2, 7, 0, 0, 0, 1, 1, 2, 2, 7, 0, 0, 1, 0, 2, 1, 3, 4, 8, 0, 0, 0, 1, 0, 2, 1, 3, 4, 8, 0, 0, 0, 0, 1, 1, 2, 1, 4, 4, 10, 0, 0, 0, 1, 1, 1, 1, 4, 2, 4, 5, 10, 0, 0, 1, 0, 1, 1, 2, 1, 4, 2, 5, 5, 11, 0, 0, 0, 1, 0, 1, 1, 2, 1, 4, 2, 5, 5, 11

Offset: 2

Henry Bottomley, Dec 06 2003

			Square array starts:
1, 0, 0, 0, 0, 0, 0, ...
1, 1, 0, 0, 1, 0, 0, ...
3, 1, 1, 0, 1, 0, 1, ...
3, 1, 1, 1, 1, 0, 1, ...
4, 2, 2, 1, 2, 0, 1, ...
4, 2, 2, 1, 2, 1, 1, ...
7, 2, 3, 1, 2, 1, 2, ...

Alois P. Heinz, Antidiagonals n = 2..142, flattened

Columns include A011371, A054861, A090616, A027868, A054861, A054896, A090617, A090618, A027868, A064458, A090619, A090620, A054896, A027868, A090621.
Cf. A090623, A090624, A115627.
Diagonal gives A011776. - Alois P. Heinz, Oct 04 2012

f:= proc(n, p) local c, k; c, k:= 0, p;
       while n>=k do c:= c+iquo(n, k); k:= k*p od; c
    end:
T:= (n, k)-> min(seq(iquo(f(n, i[1]), i[2]), i=ifactors(k)[2])):
seq(seq(T(n, 2+d-n), n=2..d), d=2..20);  # Alois P. Heinz, Oct 04 2012

f[n_, p_] := Module[{c = 0, k = p}, While[n >= k , c = c + Quotient[n, k]; k = k*p ]; c ]; t[n_, k_] := Min[ Table[ Quotient[f[n, i[[1]]], i[[2]]], {i, FactorInteger[k]}]]; Table[ Table[t[n, 2 + d - n], {n, 2, d}], {d, 2, 20}] // Flatten (* Jean-François Alcover, Oct 03 2013, translated from Alois P. Heinz's Maple program *)

For k=p prime: T(n,p) = [n/p] + [n/p^2] + [n/p^3] + .... For k = p^m a prime power: T(n,p^m) = [T(n,p)/m]. For k = b*c with b and c coprime: T(n,a*b) = min(T(n,a), T(n,b)). T(n,k) is close to, but below, n/A090624(k).

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

Original entry on oeis.org

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

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

A090630 Greatest divisor d of n! such that d=m^k with k>1.

Original entry on oeis.org

1, 1, 1, 1, 8, 8, 144, 144, 576, 5184, 518400, 518400, 2073600, 2073600, 101606400, 914457600, 14631321600, 14631321600, 526727577600, 526727577600, 52672757760000, 221225582592000, 6373403688960000, 6373403688960000, 917770131210240000, 22944253280256000000, 3877578804363264000000

Offset: 0

Reinhard Zumkeller, Dec 13 2003

nonn

a(n) is a square for all n except n = 4, 5 and 21 (Wilke, 1981). - Amiram Eldar, Jun 09 2022

Charles R Greathouse IV, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Perfect Powers.
Eric Weisstein's World of Mathematics, Factorial.
Kenneth M. Wilke, Problem 493, Pi Mu Epsilon Journal, Vol. 7, No. 4 (1981), p. 265; alternative link.
Index entries for sequences related to factorial numbers.

Cf. A011776, A060818, A060828, A251753.

f:= proc(n)
  local F,  k, d,r,s;
  F:= ifactors(n!)[2];
  r:= 1;
  for k from 2 to F[1][2] do
    r:= max(r, mul(f[1]^(k*floor(f[2]/k)),f=F))
  od:
r
end proc:
1,1,seq(f(n), n=2..100); # Robert Israel, Dec 08 2014

IsPower[n_] := If[n==1, True, GCD@@(Transpose[FactorInteger[n]][[2]])>1]; Table[Select[Divisors[n! ], IsPower][[ -1]], {n, 0, 25}]

a(n)=my(f=factor(n!),m=1); for(i=2,if(#f~,f[1,2]), m=max(factorback(concat(Mat(f[,1]), f[,2]\i*i)),m)); m \\ Charles R Greathouse IV, Dec 09 2014

a(n)= n!/A251753(n). - Robert G. Wilson v, Dec 08 2014

More terms from T. D. Noe, Oct 04 2004

A133481 a(1) = 1; for n > 1, a(n) is the least k such that k^n divides k! but k^(n+1) does not divide k!.

Original entry on oeis.org

1, 6, 15, 18, 12, 32, 24, 36, 40, 45, 48, 100, 84, 60, 154, 165, 72, 96, 80, 126, 90, 135, 286, 200, 312, 264, 168, 120, 297, 189, 160, 330, 544, 210, 144, 224, 300, 385, 396, 324, 252, 680, 350, 180, 280, 748, 572, 486, 400, 405, 315, 528, 320, 336, 450, 512, 288, 240, 715

Offset: 1

Masahiko Shin, Nov 29 2007

Least k such that A011776(k) = n.
New record highs, by index: 1, 2, 3, 4, 6, 8, 9, 10, 11, 12, 15, 16, 23, 25, 32, 33, 42, 46, 63, 66, 79, 85, 100, 119, 128, 167, 188, 201, 213, 226, 240, 256, 335, 346, 348, 352, 360, 377, 385, 414, 426, 480, 481, 494, 504, 533, 555, 596, 656, 727, 883, 926, 938, 1026, 1094, ... - Robert G. Wilson v, Feb 28 2012
First 10000 terms are 163-smooth. - David A. Corneth, Mar 15 2019

			a(7)=24 because 24^7|24! and smaller numbers than 24 do not divide their factorials 7 times.
a(2) = 6 as 6^2|6! but 6! doesn't divide 6^(2 + 1) and 6 is the least positive integer with this property. - _David A. Corneth_, Mar 15 2019

Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 251.

David A. Corneth, Table of n, a(n) for n = 1..10000 (terms 1..345 from T. D. Noe, terms 346..1150 from Robert G. Wilson v)
David A. Corneth, PARI program.
Index entries for sequences related to factorial numbers

Cf. A011776, A011777, A011778.

kdn[n_]:=Module[{k=2},While[!Divisible[k!,k^n]||Divisible[k!, k^(n+1)], k++];k]; Join[{1},Array[kdn,60,2]] (* Harvey P. Dale, Feb 27 2012 *)

a(n)=if(n<2,1,my(k=2);while(valuation(k!,k)!=n,k++);k) \\ Charles R Greathouse IV, Feb 27 2012

See Corneth link \\ David A. Corneth, Mar 15 2019

Edited by N. J. A. Sloane using material from A011777, Nov 29 2007

A011777 a(n) = least k>1 such that k^n divides k!.

Original entry on oeis.org

2, 6, 15, 18, 12, 32, 24, 36, 40, 45, 48, 100, 84, 60, 154, 165, 72, 96, 80, 126, 90, 135, 286, 200, 312, 264, 168, 120, 297, 189, 160, 330, 544, 210, 144, 224, 300, 385, 396, 324, 252, 680, 350, 180, 280, 748, 572, 486, 400, 405, 315, 528, 320, 336, 450, 512, 288, 240, 715

Offset: 1

Robert G. Wilson v

For n >= 2, least k such that A011776(k)=n.

Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 251.

See A133481 for a better version. Cf. A011776, A011778.

A074845 Numbers k such that S(k) = largest difference between consecutive divisors of k (ordered by size), where S(k) is the Kempner function (A002034).

Original entry on oeis.org

6, 8, 9, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514

Offset: 1

Jason Earls, Sep 10 2002

It appears that terms > 6 are simply given by: composite k such that k^2 doesn't divide A000254(k). - Benoit Cloitre, Mar 09 2004
It appears that A011776(a(k)) = 2. - Gionata Neri, Jul 31 2017
It appears that this sequence consists of the numbers k such that A045763(k) > 0 and k does not divide A070251(k). - Isaac Saffold, Jun 01 2018

Michael De Vlieger, Table of n, a(n) for n = 1..670 (a(n) < 10^4, from b-file at A002034).

Cf. A002034, A060681.

Select[Range@ 514, Function[n, Module[{m = 1}, While[! Divisible[m!, n], m++]; m] == Max@ Differences@ Divisors@ n]] (* Michael De Vlieger, Jul 31 2017 *)

K(n) = my(s=1); while(s!%n>0, s++); s;
dd(n) = my(vd=divisors(n)); vecmax(vector(#vd-1, k, vd[k+1] - vd[k]));
isok(n) = K(n) == dd(n); \\ Michel Marcus, Aug 03 2017

A346418 a(n) is the exponent of the largest power of n that divides the least common multiple of {1,2,...,n} (A003418). a(1) = 1.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 16 2021

nonn

			a(2) = 1 since A003418(2) = 2, and 2^1|A003418(2).
a(30) = 2 since A003418(30) = 2329089562800 = 30^2 * 2587877292, and 30^2|A003418(30).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős, Problem 10192, The American Mathematical Monthly, Vol. 99, No. 1 (1992), p. 61; An Arithmetic Function of Modest Size, solution to problem 10192 by Richard Stong, ibid., Vol. 104, No. 1 (1997), pp. 69-70.

Cf. A001221, A003418, A011776.

a[1] = 1; a[n_] := IntegerExponent[LCM @@ Range[n], n]; Array[a, 100]

a(n) = if (n==1, 1, valuation(lcm([1..n]), n)); \\ Michel Marcus, Jul 17 2021

a(n) <= omega(n), and a(n) < omega(n) whenever omega(n) > 1.

Max_{k=2..n} a(k) ~ log(n)/(log(log(n)) + o(1)) (Erdős, 1992).

cp's OEIS Frontend

A011778 Numbers k where A011776(k) grows.

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A373725 Numbers k such that A011776(k) = A011776(k+1).

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Mathematica

PARI

A090622 Square array read by antidiagonals of highest power of k dividing n! (with n,k>1).

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Maple

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

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Magma

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PARI

Sage

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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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A090630 Greatest divisor d of n! such that d=m^k with k>1.

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Maple

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A133481 a(1) = 1; for n > 1, a(n) is the least k such that k^n divides k! but k^(n+1) does not divide k!.

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