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 1-7 of 7 results.

A056630 a(n) = A055993(n) - A034444(A056627(n)).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 4, 8, 22, 22, 28, 28, 56, 88, 120, 120, 172, 172, 284, 292, 584, 584, 848, 1136, 2272, 2656, 4304, 4304, 5312, 5312, 6080, 6112, 12992, 16256, 19376, 19376, 38752, 43136, 47936, 47936, 63936, 63936, 100672, 132928, 278528, 278528
Offset: 1

Views

Author

Labos Elemer, Aug 08 2000

Keywords

Comments

Previous name, "Number of non-unitary square divisors of n!." was incorrect. See A375188 for the correct sequence with this name. - Amiram Eldar, Aug 03 2024

Examples

			example: a(10) = A055993(10) - A034444(A056627(10)) = 30 - A034444(720) = 30 - 8 = 22.

Crossrefs

Cf. A008833, A034444, A046951, A055993, A055229, A056627, A375188.

Programs

  • Mathematica
    A046951[n_] := Length[Select[Divisors[n], IntegerQ[Sqrt[#]] &]]; A008833[n_] := First[Select[Reverse[Divisors[n]], IntegerQ[Sqrt[#]] &, 1]]; A055229[n_] := With[{sf = Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n])}, GCD[sf, n/sf]]; Table[A046951[n!] - 2^(PrimeNu[Sqrt[A008833[n!]]/A055229[n!]]), {n,1,50}] (* G. C. Greubel, May 20 2017 *)
f1[p_, e_] := 1 + Floor[e/2]; f2[p_, 1] := 1; f2[p_, e_] := If[EvenQ[e], p^(e/2), p^((e-3)/2)]; ; a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n!]) - 2^PrimeNu[Times @@ f2 @@@ fct]; Array[a, 60] (* Amiram Eldar, Aug 03 2024 *)
  • PARI
    a(n) = {my(f = factor(n!)); prod(i = 1, #f~, 1 + f[i, 2]\2) - 2^omega(prod(i = 1, #f~, if(f[i, 2] == 1, 1, f[i, 1]^if(f[i, 2]%2, (f[i, 2]-3)/2, f[i, 2]/2))));} \\ Amiram Eldar, Aug 03 2024

Extensions

Incorrect name replaced with a formula by Amiram Eldar, Aug 03 2024

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

A248781 Number of integers k^4 that divide n!

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 4, 6, 6, 6, 6, 6, 6, 8, 8, 15, 15, 30, 30, 30, 30, 36, 36, 36, 48, 112, 112, 112, 112, 128, 128, 144, 216, 270, 270, 270, 270, 300, 300, 300, 300, 660, 792, 792, 792, 864, 1296, 1728, 1728, 3744, 3744, 4368, 4368, 4704, 4704, 4704
Offset: 1

Views

Author

Clark Kimberling, Oct 15 2014

Keywords

Examples

			a(10) counts these divisors of 10!:  1, 16, 81, 256, 1296, 20736.

Links

Crossrefs

Cf. A000142, A055993, A248780, A248765.

Programs

  • Mathematica
    z = 130; m = 4;
f[n_] := f[n] = FactorInteger[n!]; r[x_] := r[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]]}];
a[n_] := Apply[Times, 1 + r[v[n]]/m]
t = Table[a[n], {n, 1, z}] (* A248781 *)
  • PARI
    a(n)=c=0;d=divisors(n!);for(i=1,#d,if(ispower(d[i])&&ispower(d[i])%4==0,c++));c+1
n=1;while(n<50,print1(a(n),", ");n++) \\ Derek Orr, Oct 20 2014

A375187 Number of unitary square divisors of n!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 4, 2, 2, 8, 8, 4, 4, 4, 4, 4, 4, 8, 8, 16, 4, 4, 4, 16, 16, 16, 8, 16, 16, 32, 32, 16, 4, 16, 16, 16, 16, 16, 16, 16, 16, 8, 8, 16, 32, 128, 128, 256, 256, 128, 32, 64, 64, 256, 64, 16, 4, 16, 16, 64, 64, 64, 128, 128, 32, 64, 64, 128, 128
Offset: 0

Views

Author

Amiram Eldar, Aug 03 2024

Keywords

Comments

Unitary analog of A046951(n!) = A055993(n).

Links

Crossrefs

Cf. A000142, A046951, A055993, A056624, A375188.

Programs

  • Mathematica
    f[p_, e_] := 2^(1 - Mod[e, 2]); a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 100, 0]
  • PARI
    a(n) = vecprod(apply(x -> 1 << (1 - x%2), factor(n!)[,2]));
  • Python
    from collections import Counter
from sympy import factorint
def A375187(n): return 1<Chai Wah Wu, Aug 03 2024

Formula

a(n) = A056624(n!).

A248782 Number of integers k^5 that divide n!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 15, 30, 30, 30, 36, 36, 36, 36, 42, 56, 56, 112, 112, 112, 128, 128, 128, 128, 128, 128, 144, 270, 270, 270, 300, 300, 300, 300, 300, 300, 396, 792, 792, 792, 792, 792, 864, 864, 864, 1512
Offset: 1

Views

Author

Clark Kimberling, Oct 15 2014

Keywords

Examples

			a(12) counts these divisors of 12!:  1, 32, 243, 1024, 7776, 248832.

Links

Crossrefs

Cf. A000142, A055993, A248780, A248781, A248768.

Programs

  • Mathematica
    z = 130; m = 5;
f[n_] := f[n] = FactorInteger[n!]; r[x_] := r[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]]}];
a[n_] := Apply[Times, 1 + r[v[n]]/m]
t = Table[a[n], {n, 1, z}] (* A248782 *)
  • PARI
    a(n)=c=0;d=divisors(n!);for(i=1,#d,if(ispower(d[i])&&ispower(d[i])%5==0,c++));c+1
n=1;while(n<50,print1(a(n),", ");n++) \\ Derek Orr, Oct 20 2014

A056596 Number of nonsquare divisors of n!.

Original entry on oeis.org

0, 1, 3, 6, 14, 24, 54, 88, 148, 240, 510, 756, 1548, 2520, 3936, 5248, 10624, 14508, 29196, 40740, 60500, 95400, 191400, 242016, 338880, 529920, 674688, 912912, 1830192, 2327424, 4660224, 5523456, 7858176, 12152064, 16406592, 19576080
Offset: 1

Views

Author

Labos Elemer, Jul 21 2000

Keywords

Links

Crossrefs

A000142, A000005, A046951, A055772, A027423, A055993.

Programs

Formula

a(n) = d(n!) - A046951(n!)
a(n) = A027423(n) - A055993(n). - Amiram Eldar, Jul 16 2019

A248783 Number of integers k^6 that divide n!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 4, 6, 6, 6, 6, 8, 8, 8, 8, 8, 16, 16, 24, 30, 30, 30, 30, 36, 36, 36, 36, 36, 36, 36, 48, 56, 56, 112, 112, 112, 112, 128, 128, 128, 128, 192, 192, 216, 216, 270, 270, 270, 270, 300, 300, 300, 300, 300, 360, 396, 396
Offset: 1

Views

Author

Clark Kimberling, Oct 15 2014

Keywords

Examples

			a(16) counts these divisors of 16!:  1^6, 2^6, 2^12, 3^6, 6^6, 12^6.

Links

Crossrefs

Cf. A000142, A055993, A248780, A248781, A248782, A248771.

Programs

  • Mathematica
    z = 130; m = 6;
f[n_] := f[n] = FactorInteger[n!]; r[x_] := r[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]]}];
a[n_] := Apply[Times, 1 + r[v[n]]/m]
t = Table[a[n], {n, 1, z}] (* A248783 *)
  • PARI
    a(n)=c=0;d=divisors(n!);for(i=1,#d,if(ispower(d[i])&&ispower(d[i])%6==0,c++));c+1
n=1;while(n<50,print1(a(n),", ");n++) \\ Derek Orr, Oct 20 2014
Showing 1-7 of 7 results.

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