cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A217467 a(1) = 1; for n > 1, the maximum exponent k such that n^k divides the double factorial n!!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 2, 1, 1, 5, 2, 1, 2, 2, 1, 3, 1, 6, 2, 1, 3, 4, 1, 1, 2, 4, 1, 3, 1, 2, 6, 1, 1, 10, 2, 3, 2, 2, 1, 4, 3, 4, 2, 1, 1, 7, 1, 1, 6, 10, 3, 3, 1, 2, 2, 5, 1, 8, 1, 1, 5, 2, 4, 3, 1, 9, 5, 1, 1, 6, 3, 1
Offset: 1

Views

Author

Michel Lagneau, Oct 10 2012

Keywords

Comments

n !! is a double factorial number (see the definition in A006882).

Examples

			24^5 = 7962624 divides 24!! = 1961990553600 but 24^6 does not so a(24)=5.

Links

Crossrefs

Cf. A006882, A011776.

Programs

  • Maple
    A217467 := proc(n)
    local df,k ;
    if n = 1 then
        return 1;
    end if;
    df := doublefactorial(n) ;
    for k from 1 do
        if (df mod n^(k+1)) <> 0 then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Oct 10 2012
  • Mathematica
    Join[{1}, Table[IntegerExponent[n!!, n], {n, 2, 200}]]
  • PARI
    a(n)={my(h=(n+1)\2); if (n==1, 1, valuation(if(n%2, (2*h)!/(2^h*h!), 2^h*h!), n))} \\ Andrew Howroyd, Feb 25 2018

A060067 Largest power of n which divides n!.

Original entry on oeis.org

1, 2, 3, 4, 5, 36, 7, 64, 81, 100, 11, 248832, 13, 196, 3375, 4096, 17, 104976, 19, 160000, 9261, 484, 23, 4586471424, 15625, 676, 531441, 614656, 29, 21870000000, 31, 1073741824, 35937, 1156, 52521875, 2821109907456, 37, 1444, 59319
Offset: 1

Views

Author

Henry Bottomley, Feb 19 2001

Keywords

Examples

			a(12) = 12^5 = 248832 since 12! = 479001600 = 2^10*3^5*5^2*7*11 and 12 = 2^2*3.

Crossrefs

Cf. A011776, A000142, A060068.

Programs

  • Mathematica
    Join[{1},Table[n^IntegerExponent[n!,n],{n,2,40}]] (* Harvey P. Dale, May 06 2018 *)
  • PARI
    a(n) = if (n==1, 1, n^valuation(n!, n)); \\ Michel Marcus, Mar 23 2020

Formula

a(n) = n^A011776(n) = A000142(n)/A060068(n).

A060068 Divide n! by largest power of n which will leave the result an integer.

Original entry on oeis.org

1, 1, 2, 6, 24, 20, 720, 630, 4480, 36288, 3628800, 1925, 479001600, 444787200, 387459072, 5108103000, 20922789888000, 60988928000, 6402373705728000, 15205637551104, 5516784599040000, 2322315553259520000, 1124000727777607680000, 135277939046250
Offset: 1

Views

Author

Henry Bottomley, Feb 19 2001

Keywords

Examples

			a(12) = 1925 since 12! = 479001600 and dividing repeatedly by 12 gives 39916800, 3326400, 277200, 23100, 1925, 160.416666..., ...

Crossrefs

Cf. A000142, A011776, A060067.

Programs

  • Mathematica
    Join[{1},Table[n!/n^IntegerExponent[n!,n],{n,2,30}]] (* Harvey P. Dale, May 01 2013 *)

Formula

a(n) = n!/n^A011776(n) = A000142(n)/A060067(n).

Extensions

Offset corrected by Sean A. Irvine, Oct 23 2022

A117134 Greatest k such that n^k divides (n^2)!.

Original entry on oeis.org

3, 4, 7, 6, 17, 8, 21, 20, 24, 12, 70, 14, 32, 55, 63, 18, 80, 20, 99, 73, 48, 24, 191, 78, 56, 121, 130, 30, 224, 32, 204, 108, 72, 203, 323, 38, 80, 126, 398, 42, 293, 44, 193, 505, 96, 48, 575, 200, 312, 162, 225, 54, 485, 302, 522, 180, 120, 60, 898, 62, 128, 660, 682
Offset: 2

Views

Author

Robert Israel, Apr 26 2007

Keywords

Comments

If p is prime, a(p) = p+1, a(p^2) = floor((p^3 + p^2 + p + 1)/2).

Examples

			a(3)=4 because (3^2)! = 362880 = 3^4 * 4480 and 4480 is not divisible by 3.

References

  • Thread "100!" in rec.puzzles newsgroup, April 2007

Links

Crossrefs

Cf. A011776.

Programs

  • Maple
    seq(ordp((n^2)!,n), n=2..50);
# Alternative:
f:= proc(n) local F,m,t,v,j;
  F:= ifactors(n)[2];
  m:= infinity;
  for t in F do
    v:= add(floor(n^2/t[1]^j),j=1..ceil(log[t[1]](n^2)));
    m:= min(m,floor(v/t[2]));
  od;
  m
end proc:
map(f, [$2..100]); # Robert Israel, Feb 26 2019
  • Mathematica
    gkn[n_]:=Module[{c=(n^2)!,k},k=Floor[Log[c]/Log[n]]; While[!Divisible[ c,n^k], k--];k]; Array[gkn,70,2] (* Harvey P. Dale, Sep 14 2012 *)

A071637 Largest exponent k >=0 such that (n+1)^k divides n!.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 4, 0, 1, 2, 2, 0, 3, 0, 3, 2, 1, 0, 6, 2, 1, 3, 3, 0, 6, 0, 5, 2, 1, 4, 7, 0, 1, 2, 8, 0, 5, 0, 3, 9, 1, 0, 10, 3, 5, 2, 3, 0, 7, 4, 8, 2, 1, 0, 13, 0, 1, 9, 9, 4, 5, 0, 3, 2, 10, 0, 16, 0, 1, 8, 3, 6, 5, 0, 18, 9, 1, 0, 12, 4, 1, 2, 7, 0, 20, 6, 3, 2, 1, 4, 17, 0, 7, 8, 11
Offset: 1

Views

Author

Benoit Cloitre, Jun 25 2002

Keywords

Comments

a(A068499(n)) = 0.

Examples

			12^4 divides 11! (11!/12^4=1925) but 12^5 doesn't, hence a(11)=4.

Crossrefs

A011776(n+1) - 1.

Programs

  • Mathematica
    Table[IntegerExponent[n!,n+1],{n,500}] (* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)
  • PARI
    for(n=1,150,s=0; while(n!%(n+1)^s==0,s++); print1(s-1,","))

A098094 T(n,k) = greatest e such that k^e divides n!, 2<=k<=n (triangle read by rows).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 4, 2, 2, 1, 2, 4, 2, 2, 1, 2, 1, 7, 2, 3, 1, 2, 1, 2, 7, 4, 3, 1, 4, 1, 2, 2, 8, 4, 4, 2, 4, 1, 2, 2, 2, 8, 4, 4, 2, 4, 1, 2, 2, 2, 1, 10, 5, 5, 2, 5, 1, 3, 2, 2, 1, 5, 10, 5, 5, 2, 5, 1, 3, 2, 2, 1, 5, 1, 11, 5, 5, 2, 5, 2, 3, 2, 2, 1, 5, 1, 2, 11, 6, 5, 3, 6, 2, 3, 3, 3, 1, 5, 1, 2, 3
Offset: 2

Views

Author

Reinhard Zumkeller, Sep 14 2004

Keywords

Examples

			Array begins:
  1
  1 1
  3 1 1
  3 1 1 1
  4 2 2 1 2
  ...

Links

Crossrefs

Cf. A011371, A054861, A011776.

Programs

  • Mathematica
    T[n_, k_] := IntegerExponent[n!, k];
Table[T[n, k], {n, 2, 15}, {k, 2, n}] // Flatten (* Jean-François Alcover, Sep 15 2021 *)
  • PARI
    T(n,k) = valuation(n!, k); \\ Michel Marcus, Sep 15 2021

Formula

T(n,2) = A011371(n); T(n,3) = A054861(n) for n>2; T(n,n) = A011776(n).
Previous Showing 11-16 of 16 results.

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