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

Author

Robert G. Wilson v

Keywords

• nonn
• easy
• look
• nice

Comments

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

Examples

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

References

• 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.

Links

• 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.

Crossrefs

Cf. A011777, A011778, A133481, A000142, A074845.

Diagonal of A090622.

Cf. A175787 (primes together with 4).

Programs

Haskell

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

Maple

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

Mathematica

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 *)

PARI

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

PARI

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