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.

Showing 1-5 of 5 results.

A145642 Cubefree part of n!.

Original entry on oeis.org

1, 2, 6, 3, 15, 90, 630, 630, 210, 2100, 23100, 34650, 450450, 6306300, 28028, 7007, 119119, 2144142, 40738698, 101846745, 230945, 5080790, 116858170, 350574510, 70114902, 1822987452, 1822987452, 6380456082, 185033226378, 5550996791340
Offset: 1

Views

Author

Artur Jasinski, Oct 15 2008

Keywords

Links

Crossrefs

Cf. A000142 (n!), A050985 (cubefree part), A248762, A248763.
CF. A055204 (squarefree part of n!).

Programs

  • Mathematica
    CubefreePart[n_Integer?Positive] := Times @@ Power @@@ ({#[[1]], Mod[ #[[2]], 3]} & /@ FactorInteger[n]); Table[CubefreePart[n! ], {n, 1, 40}]
  • PARI
    a(n) = my(f=factor(n!)); f[,2] = apply(x->(x % 3), f[,2]); factorback(f); \\ Michel Marcus, Jan 06 2019
  • Python
    from operator import mul
from functools import reduce
from sympy import factorint
import math
def A145642(n):
    return 1 if n <=1 else reduce(mul,[p**(e % 3) for p,e in factorint(math.factorial(n)).items()])
# Chai Wah Wu, Feb 04 2015

Formula

a(n) = A050985(A000142(n)). - Michel Marcus, Nov 07 2013
From Amiram Eldar, Sep 01 2024: (Start)
a(n) = n! / A248762(n) = n! / A248763(n)^3.
log(a(n)) = log(3) * n + o(n) (Jakimczuk, 2017). (End)

A248780 Number of cubes that divide n!

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 6, 6, 6, 8, 8, 8, 24, 36, 36, 36, 36, 42, 112, 112, 112, 128, 192, 192, 240, 270, 270, 270, 270, 330, 792, 792, 792, 864, 864, 864, 2016, 2912, 2912, 4704, 4704, 4704, 5376, 5760, 5760, 6144, 6144, 7680, 15360, 16320, 16320, 18360
Offset: 1

Views

Author

Clark Kimberling, Oct 15 2014

Keywords

Examples

			a(9) counts these divisors of 9!:  1, 8, 27, 64, 216, 1728.

Links

Crossrefs

Cf. A000142, A055993, A061704, A248781, A248762.

Programs

  • Mathematica
    z = 130; m = 3; f[n_] := f[n] = FactorInteger[n!];
v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
a[n_] := Apply[Times, 1 + Floor[v[n]/m]]
A248780 = Table[a[n], {n, 1, z}] (* simplified by M. F. Hasler, Oct 22 2014 *)
  • PARI
    a(n)=sumdiv(n!,d,ispower(d,3))
for(n=1,50,print1(a(n),", ")) \\ Derek Orr, Oct 20 2014, simplified by M. F. Hasler, Oct 22 2014
  • PARI
    A248780(n)=prod(i=1,#n=factor(n!)[,2],1+n[i]\3) \\ M. F. Hasler, Oct 22 2014

Formula

a(n) = product_{i=1..r} 1+floor(e[i]/3), where product_{i=1..r} p[i]^e[i] is the prime factorization of n!. - M. F. Hasler, Oct 22 2014
a(n) = A061704(A000142(n)). - Michel Marcus, Mar 27 2015

A248765 Greatest k such that k^4 divides n!

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 6, 12, 12, 12, 12, 12, 12, 24, 24, 144, 144, 720, 720, 720, 720, 1440, 1440, 1440, 4320, 60480, 60480, 60480, 60480, 120960, 120960, 241920, 1209600, 3628800, 3628800, 3628800, 3628800, 7257600
Offset: 1

Views

Author

Clark Kimberling, Oct 14 2014

Keywords

Comments

Every term divides all its successors.

Examples

			a(6) = 2 because 2^4 divides 6! and if k > 2 then k^4 does not divide 6!.

Links

Crossrefs

Cf. A000142, A053164, A248762, A248764, A248766.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = FactorInteger[n!]; r[m_, x_] := r[m, x] = m*Floor[x/m];
u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
p[m_, n_] := p[m, n] = Product[u[n][[i]]^r[m, v[n]][[i]], {i, 1, Length[f[n]]}];
m = 4; Table[p[m, n], {n, 1, z}]  (* A248764 *)
Table[p[m, n]^(1/m), {n, 1, z}]   (* A248765 *)
Table[n!/p[m, n], {n, 1, z}]      (* A248766 *)
f[p_, e_] := p^Floor[e/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 30] (* Amiram Eldar, Sep 01 2024 *)
  • PARI
    a(n) = {my(f = factor(n!)); prod(i = 1, #f~, f[i, 1]^(f[i, 2]\4));} \\ Amiram Eldar, Sep 01 2024

Formula

From Amiram Eldar, Sep 01 2024: (Start)
a(n) = A053164(n!).
a(n) = (n! / A248766(n))^(1/4) = A248764(n)^(1/4).
log(a(n)) = (1/4)*n*log(n) - (2*log(2)+1)*n/4 + o(n) (Jakimczuk, 2017). (End)

A248763 Greatest k such that k^3 divides n!

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 4, 12, 12, 12, 24, 24, 24, 360, 1440, 1440, 1440, 1440, 2880, 60480, 60480, 60480, 120960, 604800, 604800, 1814400, 3628800, 3628800, 3628800, 3628800, 14515200, 479001600, 479001600, 479001600, 958003200, 958003200, 958003200
Offset: 1

Views

Author

Clark Kimberling, Oct 14 2014

Keywords

Comments

Every term divides all its successors.

Examples

			a(4) = 2 because 2^3 divides 24 and if k > 2, then k^3 > 8 does not divide 24.

Links

Crossrefs

Cf. A000142, A053150, A145642, A248762.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = FactorInteger[n!]; r[m_, x_] := r[m, x] = m*Floor[x/m];
u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
p[m_, n_] := p[m, n] = Product[u[n][[i]]^r[m, v[n]][[i]], {i, 1, Length[f[n]]}];
m = 3; Table[p[m, n], {n, 1, z}]  (* A248762 *)
Table[p[m, n]^(1/m), {n, 1, z}]   (* A248763 *)
Table[n!/p[m, n], {n, 1, z}]      (* A145642 *)
f[p_, e_] := p^Floor[e/3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 40] (* Amiram Eldar, Sep 01 2024 *)
  • PARI
    a(n) = {my(f = factor(n!)); prod(i = 1, #f~, f[i, 1]^(f[i, 2]\3));} \\ Amiram Eldar, Sep 01 2024

Formula

From Amiram Eldar, Sep 01 2024: (Start)
a(n) = A053150(n!).
a(n) = (n! / A145642(n))^(1/3) = A248762(n)^(1/3).
log(a(n)) = (1/3)*n*log(n) - (log(3)+1)*n/3 + o(n) (Jakimczuk, 2017). (End)

A248764 Greatest 4th power integer that divides n!

Original entry on oeis.org

1, 1, 1, 1, 1, 16, 16, 16, 1296, 20736, 20736, 20736, 20736, 20736, 20736, 331776, 331776, 429981696, 429981696, 268738560000, 268738560000, 268738560000, 268738560000, 4299816960000, 4299816960000, 4299816960000, 348285173760000, 13379723235164160000
Offset: 1

Views

Author

Clark Kimberling, Oct 14 2014

Keywords

Comments

Every term divides all its successors.

Examples

			a(6) = 16 because 16 divides 6! and if k > 2 then k^4 does not divide 6!.

Links

Crossrefs

Cf. A000142, A008835, A248762, A248765, A248766.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = FactorInteger[n!]; r[m_, x_] := r[m, x] = m*Floor[x/m];
u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
p[m_, n_] := p[m, n] = Product[u[n][[i]]^r[m, v[n]][[i]], {i, 1, Length[f[n]]}];
m = 4; Table[p[m, n], {n, 1, z}]  (* A248764 *)
Table[p[m, n]^(1/m), {n, 1, z}]   (* A248765 *)
Table[n!/p[m, n], {n, 1, z}]      (* A248766 *)
f[p_, e_] := p^(4*Floor[e/4]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 30] (* Amiram Eldar, Sep 01 2024 *)
  • PARI
    a(n) = {my(f = factor(n!)); prod(i = 1, #f~, f[i, 1]^(4*(f[i, 2]\4)));} \\ Amiram Eldar, Sep 01 2024

Formula

a(n) = n!/A248766(n).
From Amiram Eldar, Sep 01 2024: (Start)
a(n) = A008835(n!).
a(n) = A248765(n)^4. (End)
Showing 1-5 of 5 results.

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