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 1-3 of 3 results.

A056629 a(n) = A034444(A056627(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 4, 4, 4, 8, 8, 8, 8, 16, 8, 8, 8, 8, 8, 16, 8, 16, 16, 16, 16, 32, 32, 64, 64, 64, 64, 64, 32, 64, 64, 64, 64, 128, 64, 64, 64, 64, 64, 128, 128, 256, 256, 256, 256, 256, 128, 256, 256, 256, 256, 256, 128, 256, 256, 256, 256, 512, 512, 512, 512, 512
Offset: 1

Views

Author

Labos Elemer, Aug 08 2000

Keywords

Comments

Previous name, "Number of unitary square divisors of n!." was incorrect. See A375187 for the correct sequence with this name. - Amiram Eldar, Aug 03 2024

Examples

			a(10) = A034444(A056627(10)) = A034444(720) = 8.

Crossrefs

Cf. A008833, A034444, A055229, A056627, A375187.

Programs

  • Mathematica
    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[2^(PrimeNu[Sqrt[A008833[n!]]/A055229[n!]]), {n, 1, 50}] (* G. C. Greubel, May 19 2017 *)
f[p_, 1] := 1; f[p_, e_] := If[EvenQ[e], p^(e/2), p^((e-3)/2)]; a[1] = 1; a[n_] := 2^PrimeNu[Times @@ f @@@ FactorInteger[n!]]; Array[a, 66] (* Amiram Eldar, Aug 03 2024 *)
  • PARI
    a(n) = {my(f = factor(n!)); 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

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

A056628 a(n) = A056623(n!).

Original entry on oeis.org

1, 1, 1, 1, 1, 144, 144, 144, 1296, 518400, 518400, 230400, 230400, 2822400, 9144576, 146313216, 146313216, 21069103104, 21069103104, 52672757760000, 119439360000, 3613040640000, 3613040640000, 18730002677760000, 468250066944000000, 19783565328384000000, 19783565328384000000
Offset: 1

Views

Author

Labos Elemer, Aug 08 2000

Keywords

Comments

Previous name "Largest unitary square divisor of n!" was incorrect. See A374988 for the correct sequence with this name. - Amiram Eldar, Jul 26 2024

Examples

			a(12) = A056623(12!) = A008833(12!)/A055229(12!)^2 = 2073600/3^2 = 230400.

Crossrefs

Cf. A055772, A055071, A055230, A000188, A008833, A055229, A056623, A056627, A374988.

Programs

Formula

a(n) = A055071(n)/A055230(n)^2 = A008833(n!)/A055229(n!)^2.
a(n) = A056623(n!). - Michel Marcus, Aug 16 2020
a(n) = A056627(n)^2. - Amiram Eldar, Jul 08 2024

Extensions

More terms from Michel Marcus, Aug 16 2020
Incorrect name replaced with a formula by Amiram Eldar, Jul 26 2024
Showing 1-3 of 3 results.

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