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

1,6

Comments

From Stefano Spezia, Nov 08 2018: (Start)

It appears that a(n) = 1 when n = 4 or a prime number (see A175787).

It appears that a(n) = 2 for n in A074845. (End)

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.

J. 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)

P. Shui, A footnote on the number of times n goes into n!, The Mathematical Gazette 93:528 (2009), pp. 492-495.

Eric Weisstein's World of Mathematics, Factorial

Index entries for sequences related to factorial numbers

Example

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

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 *)

Prog

(Haskell)
a011776 1 = 1
a011776 n = length $
   takeWhile ((== 0) . (mod (a000142 n))) $ iterate (* n) n
-- Reinhard Zumkeller, Sep 01 2012
(PARI) a(n)=valuation(n!, n) \\ 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

Crossrefs

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

Diagonal of A090622.

Cf. A175787 (primes together with 4).

Sequence in context: A324386 A233390 A324114 * A345182 A297791 A098965

Adjacent sequences: A011773 A011774 A011775 * A011777 A011778 A011779

Keyword

nonn,easy,look,nice

Author

Robert G. Wilson v

Status

approved