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.

A056192 a(n) = n divided by its characteristic cube divisor A056191.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 4, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 9, 73, 74, 75
Offset: 1

Views

Author

Labos Elemer, Aug 02 2000

Keywords

Comments

Different from A056552: e.g. a(16) = 16, while A056552(16) = 2.

Links

Crossrefs

Cf. A000188, A008833, A007913, A055229, A055231, A056552, A056191.

Programs

  • Mathematica
    f[p_, e_] := If[EvenQ[e], p^e, If[e == 1, p, p^(e - 3)]]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 06 2020 *)

Formula

a(n) = n/A055229(n)^3 = n/g^3=n/ggg and n=(LL)*(ggg)*f=L^2*g^3*f=LL*a(n)^3*f, so n=L^2*(g*3)*f, where L=A000188(n)/A055229(n), f=A055231(n), g=A055231(n).
Multiplicative with a(p^e)=p^e for even e, a(p)=p, a(p^e)=p^(e-3) for odd e>1. - Vladeta Jovovic, Apr 30 2002
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4 + 1/p^6 - 1/p^7) = 0.4462648652... . - Amiram Eldar, Nov 13 2022

A056194 Characteristic cube divisor of n!: a(n) = A056191(n!).

Original entry on oeis.org

1, 1, 1, 8, 8, 1, 1, 8, 8, 1, 1, 27, 27, 216, 1000, 1000, 1000, 125, 125, 1, 9261, 74088, 74088, 343, 343, 2744, 74088, 216, 216, 125, 125, 1000, 35937000, 4492125, 12326391, 12326391, 12326391, 98611128, 8024024008, 125375375125
Offset: 1

Views

Author

Labos Elemer, Aug 02 2000

Keywords

Links

Crossrefs

Cf. A000142, A055229, A000188, A008833, A007913, A055231, A055230, A055772, A055071, A055204, A055773 A056552, A056195.

Programs

  • Mathematica
    f[p_, e_] := If[OddQ[e] && e > 1, p^3, 1]; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 40] (* Amiram Eldar, Sep 06 2020 *)

Formula

a(n) = A056191(A000142(n)). - Amiram Eldar, Sep 06 2020

A056201 Characteristic cube divisor (A056191) of central binomial coefficient (A001405).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 8, 27, 216, 8, 1, 1, 1, 27, 27, 1, 1, 1, 8, 1, 1, 1, 8, 1, 8, 216, 27, 1, 1, 1, 8, 27, 216, 8, 1, 1, 8, 8, 1, 8, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 8, 1, 8, 8, 1, 1, 1, 1, 8, 27, 216, 216, 27, 1, 8, 8, 1, 8, 1, 27, 216
Offset: 1

Views

Author

Labos Elemer, Aug 02 2000

Keywords

Examples

			n=14, binomial(14,8) = 3432 = 2*2*2*3*11*13 so g(3432)=2, thus a(14)=8.

Crossrefs

Cf. A056056-A056060, A056202.
Equals A056059^3.

Formula

a(n) = A056059(n)^3 = g^3 and binomial(n, floor(n/2)) = a(n)^3 * L^2 * A056060(n), where L = A056056(n)/A056059(n).

A366126 The largest unitary divisor of n that is a cube.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Sep 30 2023

Keywords

Comments

First differs from A056191 at n = 32.

Links

Crossrefs

Cf. A001221, A001222, A000578, A056191, A350388, A366124, A366125.

Programs

  • Mathematica
    f[p_, e_] := If[Divisible[e, 3], p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] % 3, 1, f[i,1]^f[i,2]));}

Formula

Multiplicative with a(p^e) = p^e if e is divisible by 3 and 1 otherwise.
a(n) = n if and only if n is a positive cube (A000578 \ {0}).
A001221(a(n)) = A366124(n).
A001222(a(n)) = A366125(n).
Sum_{k=1..n} a(k) ~ c * n^(4/3), where c = (1/4) * Product_{p prime} (1 + (p^(1/3) + p^(5/3))/(1 + p + p^2 + p^3)) = 0.61488587249270755696... .

A368167 The largest unitary divisor of n that is a cubefull exponentially odd number (A335988).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 14 2023

Keywords

Comments

First differs from A056191 and A366126 at n = 32, and from A367513 at n = 64.
Also, the largest exponentially odd unitary divisor of the powerful part on n.
Also, the powerful part of the largest exponentially odd unitary divisor of n.

Links

Crossrefs

Cf. A057521, A077610, A335275, A335988, A350389, A368168, A368169.
Cf. A056191, A366126, A367513.

Programs

  • Mathematica
    f[p_, e_] := If[e == 1 || EvenQ[e], 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1 || !(f[i, 2]%2), 1, f[i, 1]^f[i, 2]));}

Formula

Multiplicative with a(p^e) = p^e if e is odd that is larger than 1, and 1 otherwise.
a(n) = A350389(A057521(n)).
a(n) = A057521(A350389(n)).
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= n, with equality if and only if n is in A335988.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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