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.

A382888 The squarefree kernel of the n-th cubefree number.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 3, 10, 11, 6, 13, 14, 15, 17, 6, 19, 10, 21, 22, 23, 5, 26, 14, 29, 30, 31, 33, 34, 35, 6, 37, 38, 39, 41, 42, 43, 22, 15, 46, 47, 7, 10, 51, 26, 53, 55, 57, 58, 59, 30, 61, 62, 21, 65, 66, 67, 34, 69, 70, 71, 73, 74, 15, 38, 77, 78, 79, 82
Offset: 1

Views

Author

Amiram Eldar, Apr 07 2025

Keywords

Links

Crossrefs

Cf. A002117, A004709, A005117, A007947, A371188.
Similar sequences: A382889, A382890, A382891.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n]}, If[AllTrue[f[[;; , 2]], # < 3 &], Times @@ f[[;; , 1]], Nothing]]; Array[s, 100]
  • PARI
    list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 3, print1(vecprod(f[, 1]), ", ")));}

Formula

a(n) = A007947(A004709(n)).
a(n) = A004709(n)/sqrt(A382889(n)) = A004709(n)/A382890(n).
a(n) = sqrt(A004709(n)*A382891(n)).
a(A371188(n)) = A005117(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2. where c = zeta(3)^2 * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 0.98875409459226057523... .

A382889 The largest square dividing the n-th cubefree number.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 4, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 25, 1, 4, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 49, 25, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 1, 1, 25, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 1, 49, 9, 100
Offset: 1

Views

Author

Amiram Eldar, Apr 07 2025

Keywords

Comments

Also, the powerful part of the n-th cubefree number.
All the terms are squares of squarefree numbers (A062503).

Links

Crossrefs

Cf. A002117, A004709, A008833, A057521, A062503, A371188 (positions of 1's).
Similar sequences: A382888, A382890, A382891.

Programs

  • Mathematica
    f[p_, e_] := p^If[e == 1, 0, 2]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]
  • PARI
    list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 3, print1(prod(i = 1, #f~, f[i, 1]^if(f[i, 2] == 1, 0, 2)), ", ")));}

Formula

a(n) = A008833(A004709(n)).
a(n) = A057521(A004709(n)).
a(n) = A382890(n)^2.
a(n) = A004709(n)/A382891(n).
a(n) = (A004709(n)/A382888(n))^2.
a(A371188(n)) = 1.
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = zeta(3)^(3/2) * Product_{p prime} (1 + 1/p^(3/2) - 1/p^2 - 1/p^(5/2)) = 1.48513488319516447978... .

A382890 The square root of the largest square dividing the n-th cubefree number.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 5, 1, 2, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 7, 5, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 7, 3, 10, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Apr 07 2025

Keywords

Comments

The product of the non-unitary prime divisors of the n-th cubefree number.
Also, the square root of the powerful part of the n-th cubefree number.
All the terms are squarefree.

Links

Crossrefs

Cf. A000188, A004709, A005117, A057521, A371188 (positions of 1's).
Similar sequences: A382888, A382889, A382891.

Programs

  • Mathematica
    f[p_, e_] := p^If[e == 1, 0, 1]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]
  • PARI
    list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 3, print1(prod(i = 1, #f~, f[i, 1]^if(f[i, 2] == 1, 0, 1)), ", ")));}

Formula

a(n) = A000188(A004709(n)).
a(n) = sqrt(A382889(n)).
a(n) = A004709(n)/A382888(n).
a(n) = sqrt(A004709(n)/A382891(n)).
a(A371188(n)) = 1.

A382891 The powerfree part of the n-th cubefree number.

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 1, 10, 11, 3, 13, 14, 15, 17, 2, 19, 5, 21, 22, 23, 1, 26, 7, 29, 30, 31, 33, 34, 35, 1, 37, 38, 39, 41, 42, 43, 11, 5, 46, 47, 1, 2, 51, 13, 53, 55, 57, 58, 59, 15, 61, 62, 7, 65, 66, 67, 17, 69, 70, 71, 73, 74, 3, 19, 77, 78, 79, 82, 83
Offset: 1

Views

Author

Amiram Eldar, Apr 07 2025

Keywords

Comments

Also, the squarefree part of the n-th cubefree number.

Links

Crossrefs

Cf. A002117, A004709, A005117, A055231, A007913, A371188.
Similar sequences: A382888, A382889, A382890.

Programs

  • Mathematica
    f[p_, e_] := p^If[e == 1, 1, 0]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]
  • PARI
    list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 3, print1(prod(i = 1, #f~, f[i, 1]^if(f[i, 2] == 1, 1, 0)), ", ")));}

Formula

a(n) = A055231(A004709(n)).
a(n) = A007913(A004709(n)).
a(n) = A004709(n)/A382889(n) = A004709(n)/A382890(n)^2.
a(n) = A382888(n)^2/A004709(n).
a(A371188(n)) = A005117(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(3)^2 * Product_{p prime} (1 - 1/p^2 + 1/p^4 - 1/p^5) = 0.92517253037215590197... .

A374460 Indices of the nonsquarefree terms in the sequence of exponentially odd numbers (A268335).

Original entry on oeis.org

7, 18, 20, 24, 31, 39, 41, 63, 69, 74, 86, 89, 91, 97, 98, 109, 115, 121, 131, 135, 154, 161, 167, 174, 177, 179, 189, 194, 200, 211, 212, 223, 234, 243, 244, 249, 250, 265, 266, 268, 273, 290, 296, 302, 314, 325, 328, 338, 343, 348, 350, 366, 367, 373, 382, 388, 393
Offset: 1

Views

Author

Amiram Eldar, Jul 09 2024

Keywords

Comments

The asymptotic density of this sequence is 1 - A059956 / A065463 = 0.13700925215474602945... .

Examples

			The first 7 exponentially odd numbers are 1, 2, 3, 5, 6, 7, and 8. A268335(7) = 8 = 3^3 is the least nonsquarefree term. Therefore a(1) = 7.

Links

Crossrefs

Cf. A005117, A059956, A065463, A268335.
Similar sequences: A361936, A363189, A371186, A371188.

Programs

  • Mathematica
    Position[Select[Range[120], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &], _?(!SquareFreeQ[#] &), Heads -> False] // Flatten
  • PARI
    isexpodd(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!(e[i] % 2), return(0))); 1;}
lista(kmax) = {my(f, c = 0); for(k = 1, kmax, if(isexpodd(k), c++; if(!issquarefree(k), print1(c, ", "))));}

Formula

A268335(a(n)) = A374459(n).
Showing 1-5 of 5 results.

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