A133481 -id:A133481 - cp's OEIS frontend

A011776 a(1) = 1; for n > 1, a(n) is defined by the property that n^a(n) divides n! but n^(a(n)+1) does not.

1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 5, 1, 2, 3, 3, 1, 4, 1, 4, 3, 2, 1, 7, 3, 2, 4, 4, 1, 7, 1, 6, 3, 2, 5, 8, 1, 2, 3, 9, 1, 6, 1, 4, 10, 2, 1, 11, 4, 6, 3, 4, 1, 8, 5, 9, 3, 2, 1, 14, 1, 2, 10, 10, 5, 6, 1, 4, 3, 11, 1, 17, 1, 2, 9, 4, 7, 6, 1, 19, 10, 2, 1, 13, 5, 2, 3, 8, 1, 21

Offset: 1

Robert G. Wilson v

From Stefano Spezia, Nov 08 2018: (Start)
It appears that for n > 1 a(n) = 1 iff n = 4 or a prime number (see A175787).
It appears that a(n) = 2 iff n is in A074845. (End)
Since a prime p is coprime to all positive integers less than p, a(p)=1. - Robert D. Rosales, Jun 17 2024
If n > 4 is composite then a(n) > 1. Proof: 1) If n is not a square of a prime, then n has a divisor d such that 1 < d < n/d < n, so d, n/d and n appear as different factors in n!, n^2 | n!, and therefore a(n) >= 2. 2) If n = p^2 is a square of a prime, then p, 2*p and p^2 appear as different factors in n! when p > 2, therefore a(n) >= 2 if n != 4. - Amiram Eldar, Jul 06 2024

			12^5 divides 12! but 12^6 does not so a(12) = 5.

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.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 251.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Nick MacKinnon, How many times does n go into n!, The Mathematical Gazette, Vol. 70, No. 453 (1986), pp. 203-205.
Peter Shui, A footnote on the number of times n goes into n!, The Mathematical Gazette, Vol. 93, No. 528 (2009), pp. 492-495.
Eric Weisstein's World of Mathematics, Factorial.
Index entries for sequences related to factorial numbers.

Cf. A011777, A011778, A133481, A000142, A074845.
Diagonal of A090622.
Cf. A175787 (primes together with 4).

a011776 1 = 1
a011776 n = length $
   takeWhile ((== 0) . (mod (a000142 n))) $ iterate (* n) n
-- Reinhard Zumkeller, Sep 01 2012

a := []; for n from 2 to 200 do i := 0: while n! mod n^i = 0 do i := i+1: od: a := [op(a),i-1]; od: a;
# second Maple program:
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:
a:= n-> min(seq(iquo(f(n, i[1]), i[2]), i=ifactors(n)[2])): a(1):=1:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 04 2012

Do[m = 1; While[ IntegerQ[ n!/n^m], m++ ]; Print[m - 1], {n, 1, 100} ]
HighestPower[n_,p_] := Module[{r,s=0,k=1}, While[r=Floor[n/p^k]; r>0, s=s+r; k++ ];s]; SetAttributes[HighestPower,Listable]; Join[{1}, Table[{p,e}=Transpose[FactorInteger[n]]; Min[Floor[HighestPower[n,p]/e]], {n,2,100}]] (* T. D. Noe, Oct 01 2008 *)
Join[{1},Table[IntegerExponent[n!,n],{n,2,500}]] (* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)
f[n_, p_] := Module[{c=0, k=p}, While[n >= k , c = c + Quotient[n, k]; k = k*p ]; c]; a[1]=1; a[n_] := Min[ Table[ Quotient[f[n, i[[1]]], i[[2]]], {i, FactorInteger[n] }]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 03 2013, after Alois P. Heinz's Maple program *)

a(n)=if(n>1, valuation(n!,n), 1); \\ Charles R Greathouse IV, Apr 10 2014

vp(n,p)=my(s); while(n\=p, s+=n); s
a(n)=if(n==1,return(1)); my(f=factor(n)); vecmin(vector(#f~, i, vp(n,f[i,1])\f[i,2])) \\ Charles R Greathouse IV, Apr 10 2014

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.

A011778 Numbers k where A011776(k) grows.

Original entry on oeis.org

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

A217672 Largest k such that k^n divides k!!

Original entry on oeis.org

1, 8, 16, 36, 24, 32, 60, 72, 80, 48, 135, 200, 168, 120, 297, 189, 144, 96, 160, 252, 180, 270, 539, 400, 405, 315, 336, 240, 594, 378, 192, 660, 525, 420, 288, 448, 600, 770, 320, 648, 504, 1001, 675, 360, 560, 1496, 1125, 972, 800, 810, 630, 1056, 1300, 384

Offset: 1

Michel Lagneau, Oct 10 2012

nonn

Largest k such that A217467(k)=n.

			a(5)=24 because 24^5 divides 24!! but 24^6 does not divide 24!!.

Cf. A006882, A133481, A217467.

kdn[n_]:=Module[{k=2}, While[!Divisible[k!!, k^n]||Divisible[k!!, k^(n+1)], k++]; k]; Join[{1}, Array[kdn, 60, 2]]

A306772 a(n) is the least number k such that k! is divisible by (k+1)^n but not by (k+1)^(n+1).

Original entry on oeis.org

1, 5, 14, 17, 11, 31, 23, 35, 39, 44, 47, 99, 83, 59, 153, 164, 71, 95, 79, 125, 89, 134, 285, 199, 311, 263, 167, 119, 296, 188, 159, 329, 543, 209, 143, 223, 299, 384, 395, 323, 251, 679, 349, 179, 279, 747, 571, 485, 399, 404, 314, 527, 319, 335, 449, 511, 287, 239, 714

Offset: 0

Jinyuan Wang, Mar 09 2019

nonn

k+1 is not a prime.
a(n) + 1 is 17-smooth in DATA. - David A. Corneth, Mar 15 2019
But fails at n 99, 114, 125, 127, 130, 135, 143, 146, ... - Michel Marcus, Apr 30 2019

			For n = 1, 1! = 1 is not divisible by 2, 2! = 2 is not divisible by 3, 3! = 6 is not divisible by 4, 4! = 24 is not divisible by 5, and 5! = 120 is divisible by 6 but not 36. Therefore a(1) = 5. - _Michael B. Porter_, Apr 21 2019

Cf. A061768, A082482, A133481, A240751.

Array[Block[{k = 1}, While[Nand[Mod[k!, (k + 1)^#] == 0, Mod[k!, (k + 1)^(# + 1)] != 0], k++]; k] &, 58] (* Michael De Vlieger, Mar 11 2019 *)

a(n) = {my(k=1); while((k! % (k+1)^n) || !(k! % (k+1)^(n+1)), k++); k; }

a(n) = A133481(n+1) - 1.
a(n) >= A061768(n).
If n = floor((p^j-1)/(j*(p-1)))-1, a(n) <= p^j-1 for prime p. For example, (p = 2), a(n) <= 2^j-1 for n = floor((2^j-1)/j)-1 (A082482(j)-1).

cp's OEIS Frontend

A011776 a(1) = 1; for n > 1, a(n) is defined by the property that n^a(n) divides n! but n^(a(n)+1) does not.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

A011778 Numbers k where A011776(k) grows.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions

A217672 Largest k such that k^n divides k!!

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A306772 a(n) is the least number k such that k! is divisible by (k+1)^n but not by (k+1)^(n+1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula