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.

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A362148 Numbers that are neither cubefree nor cubefull.

Original entry on oeis.org

24, 40, 48, 54, 56, 72, 80, 88, 96, 104, 108, 112, 120, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 224, 232, 240, 248, 250, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336, 344, 351, 352, 360, 368, 375, 376, 378, 384, 392, 400
Offset: 1

Views

Author

Bernard Schott, Apr 09 2023

Keywords

Comments

In fact, every cubefull number > 1 is noncubefree, but the converse is false.
This sequence = A046099 \ A036966 and lists these counterexamples.
Numbers k such that for some primes p and q, k is divisible by p^3*q but not by q^3. - Robert Israel, Apr 28 2023
The asymptotic density of this sequence is 1 - 1/zeta(3) = 0.168092... - Charles R Greathouse IV, Apr 28 2023
From Amiram Eldar, Sep 17 2023: (Start)
Numbers k such that A360539(k) > 1 and A360540(k) > 1.
Equivalently, numbers that have in their prime factorization at least one exponent that is smaller than 3 and at least one exponent that is larger than 2. (End)

Examples

			24 = 2^3 * 3 is noncubefree as it is divisible by the cube 2^3, but it is not cubefull because 3 divides 24 but 3^3 does not divide 24, hence 24 is a term.
648 = 2^4 * 3^3 is noncubefree as it is divisible by the cube 3^3, but it is also cubefull because primes 2 and 3 divide 648, and 2^3 and 3^3 divide also 648, so 648 is not a term.

Links

Crossrefs

Intersection of A046099 (not cubefree) and A362147 (not cubefull)
Cf. A004709 (cubefree), A036966 (cubefull), A360539, A360540.

Programs

  • Maple
    filter:= proc(n) local F;
F:= ifactors(n)[2][..,2];
  min(F) < 3 and max(F) >= 3
end proc:
select(filter, [$1..400]); # Robert Israel, Apr 28 2023
  • Mathematica
    Select[Range[500], Min[(e = FactorInteger[#][[;; , 2]])] < 3 && Max[e] > 2 &] (* Amiram Eldar, Apr 09 2023 *)
  • PARI
    isok(k) = (k>1) && (vecmax(factor(k)[, 2])>2) && (vecmin(factor(k)[, 2])<=2); \\ Michel Marcus, Apr 19 2023

Formula

Equals A362147 \ A004709.
Sum_{n>=1} 1/a(n) = 1 + zeta(s) - zeta(s)/zeta(3*s) - Product_{p prime}(1 + 1/(p^(2*s)*(p^s-1))), s > 1. - Amiram Eldar, Sep 17 2023

A360541 a(n) is the least number k such that k*n is a cubefull number (A036966).

Original entry on oeis.org

1, 4, 9, 2, 25, 36, 49, 1, 3, 100, 121, 18, 169, 196, 225, 1, 289, 12, 361, 50, 441, 484, 529, 9, 5, 676, 1, 98, 841, 900, 961, 1, 1089, 1156, 1225, 6, 1369, 1444, 1521, 25, 1681, 1764, 1849, 242, 75, 2116, 2209, 9, 7, 20, 2601, 338, 2809, 4, 3025, 49, 3249, 3364
Offset: 1

Views

Author

Amiram Eldar, Feb 11 2023

Keywords

Links

Crossrefs

Cf. A036966, A329376, A356193, A360539.

Programs

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

Formula

a(n) = 1 if and only if n is cubefull number (A036966).
a(n) = A356193(n)/n.
a(n) = A360539(n)^2/A329376(n)^3.
Multiplicative with a(p^e) = p^(max(e, 3) - e), i.e., a(p) = p^2, a(p^2) = p, and a(p^e) = 1 for e >= 3.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(2-s) - p^(-s) - p^(2-2*s) + p^(1-2*s) - p^(1-3*s) + p^(-3*s)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 2/p^5 - 1/p^6 - 1/p^8 + 2/p^9 - 1/p^10) = 0.2078815423... .

A385007 The largest unitary divisor of n that is a biquadratefree number (A046100).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 3, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69
Offset: 1

Views

Author

Amiram Eldar, Jun 15 2025

Keywords

Comments

First differs from A053165 at n = 32 = 2^5: a(32) = 1 while A053165(32) = 2.
First differs from A383764 at n = 32 = 2^5: a(32) = 1 while A383764(32) = 32.
Equivalently, a(n) is the least divisor d of n such that n/d is a 4-full number (A036967).

Links

Crossrefs

Cf. A013661, A036967, A046100, A053165, A058035, A077610, A383764.
The largest unitary divisor of n that is: A000265 (odd), A006519 (power of 2), A055231 (squarefree), A057521 (powerful), A065330 (5-rough), A065331 (3-smooth), A350388 (square), A350389 (exponentially odd), A360539 (cubefree), A360540 (cubefull), A366126 (cube), A367168 (exponentially 2^n), this sequence (biquadratefree).

Programs

  • Mathematica
    f[p_, e_] := If[e < 4, 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] < 4, f[i, 1]^f[i, 2], 1)); }

Formula

a(n) = 1 if and only if n is a 4-full number (A036967).
a(n) = n if and only if n is a biquadratefree number (A046100).
Multiplicative with a(p^e) = p^e if e <= 3, and 1 otherwise.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(1-s) - p^(-s) + p^(2-2*s) - p^(1-2*s) - p^(2-3*s) + p^(3-3*s) - p^(3-4*s) + p^(-4*s)).
Sum_{k=1..n} a(k) ~ c * zeta(2) * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^6 + 1/p^8 - 1/p^9) = 0.56331392082909224894... .

A375847 The maximum exponent in the prime factorization of the largest unitary cubefree divisor of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 1, 0, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 0, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 0, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Aug 31 2024

Keywords

Links

Crossrefs

Cf. A004709, A036966, A038109, A051903, A330596, A337050, A360539.
Cf. A007424 (analogous with the largest cubefree divisor, for n >= 2).

Programs

  • Mathematica
    a[n_] := Max[Join[{0}, Select[FactorInteger[n][[;; , 2]], # <= 2 &]]]; a[1] = 0; Array[a, 100]
  • PARI
    a(n) = {my(e = select(x -> x <= 2, factor(n)[,2])); if(#e == 0, 0, vecmax(e));}

Formula

a(n) = A051903(A360539(n)).
a(n) = 0 if and only if n is cubefull (A036966).
a(n) = 1 if and only if n is in A337050 \ A036966.
a(n) = 2 if and only if n is in A038109.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 - A330596 = 1.25146474031763643535... .
Previous Showing 11-14 of 14 results.

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