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 1-6 of 6 results.

A365488 The number of divisors of the smallest number whose cube is divisible by n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 3, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 6, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 3, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 4, 8, 2, 6, 3, 4, 2, 8, 4, 4, 4
Offset: 1

Views

Author

Amiram Eldar, Sep 05 2023

Keywords

Comments

First differs from A365171 at n = 32.
The number of divisors of the smallest cube divisible by n, A053149(n), is A365489(n).

Links

Crossrefs

Cf. A000005, A005117, A019555, A053149, A365171, A365489, A365498.

Programs

  • Mathematica
    f[p_, e_] := Ceiling[e/3] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
With[{c=Range[200]^3},Table[DivisorSigma[0,Surd[SelectFirst[c,Mod[#,n]==0&],3]],{n,90}]] (* Harvey P. Dale, Sep 15 2024 *)
  • PARI
    a(n) = vecprod(apply(x -> (x-1)\3 + 2, factor(n)[, 2]));

Formula

a(n) = A000005(A019555(n)).
Multiplicative with a(p^e) = ceiling(e/3) + 1.
a(n) <= A000005(n) with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s) * zeta(3*s) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)).
From Vaclav Kotesovec, Sep 06 2023: (Start)
Dirichlet g.f.: zeta(s)^2 * zeta(3*s) * Product_{p prime} (1 - 1/p^(2*s) - 1/p^(3*s) + 1/p^(4*s)).
Let f(s) = Product_{p prime} (1 - 1/p^(2*s) - 1/p^(3*s) + 1/p^(4*s)).
Sum_{k=1..n} a(k) ~ zeta(3) * f(1) * n * (log(n) + 2*gamma - 1 + 3*zeta'(3)/zeta(3) + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085026313185459506482223745141452711510108346133288...,
f'(1) = f(1) * Sum_{p prime} (-4 + 3*p + 2*p^2) * log(p) / (1 - p - p^2 + p^4) = f(1) * 1.4525924794451595590371439593828547341482465114411929136723476679...
and gamma is the Euler-Mascheroni constant A001620. (End)

A365173 The number of divisors d of n such that gcd(d, n/d) is an exponentially odd number (A268335).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 5, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4
Offset: 1

Views

Author

Amiram Eldar, Aug 25 2023

Keywords

Comments

First differs from A252505 at n = 64.
The sum of these divisors is A365174(n).

Links

Crossrefs

Cf. A004524, A046101, A005117, A252505, A268335, A350390, A325837, A365171, A365174.

Programs

  • Mathematica
    f[p_, e_] := Floor[(e + 5)/4] + Floor[(e + 6)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecprod(apply(x -> (x+5)\4 + (x+6)\4, factor(n)[, 2]));
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 + X^2 - X^4)/((1 - X)^2*(1 + X^2)))[n], ", ")) \\ Vaclav Kotesovec, Jan 20 2024

Formula

Multiplicative with a(p^e) = floor((e+5)/4) + floor((e+6)/4) = A004524(e+5).
a(n) <= A000005(n), with equality if and only if n is not a biquadrateful number (A046101).
a(n) >= A034444(n), with equality if and only if n is squarefree (A005117).
a(n) == 1 (mod 2) if and only if n is a square of an exponentially odd number (i.e., a number whose prime factorization include only exponents e such that e == 2 (mod 4)).
From Vaclav Kotesovec, Jan 20 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/(p^(2*s)*(1 + p^(2*s)))).
Let f(s) = Product_{p prime} (1 - 1/(p^(2*s)*(1 + p^(2*s)))).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/(p^2*(1 + p^2))) = 0.937494282731300250789438325050116436995101826036120273493270589183132928...,
f'(1) = f(1) * Sum_{p prime} (4*p^2 + 2) * log(p) / (p^6 + 2*p^4 - 1) = f(1) * 0.192452062257404507109731932640803706644036700262364333369815000973104583...
and gamma is the Euler-Mascheroni constant A001620. (End)

A369310 The number of divisors d of n such that gcd(d, n/d) is a powerful number.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 4, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 6, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 5, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 4, 8, 2, 6, 3, 4, 2, 8, 4, 4, 4
Offset: 1

Views

Author

Amiram Eldar, Jan 19 2024

Keywords

Comments

First differs from A365488 at n = 32, and from A365171 at n = 64.

Links

Crossrefs

Cf. A000005, A001620, A001694, A005117, A034444, A046100, A365171, A365488, A369309.

Programs

  • Mathematica
    f[p_, e_] := If[e <= 3, 2, e - 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecprod(apply(x -> if(x <= 3, 2, x-1), factor(n)[, 2]));
  • Python
    from math import prod
from sympy import factorint
def A369310(n): return prod(2 if e<=2 else e-1 for e in factorint(n).values()) # Chai Wah Wu, Jan 19 2024

Formula

Multiplicative with a(p^e) = 2 if e <= 3, and e-1 otherwise.
a(n) >= A034444(n), with equality if and only if n is biquadratefree (A046100).
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s)^2 * f(s), where f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(4*s)).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 2/p^2 + 1/p^4) = 0.66922021803510257394...,
f'(1)/f(1) = 2 * Sum_{p prime} (p^2-2) * log(p) / (p^4 - p^2 + 1) = 0.81150060034711480230..., and gamma is Euler's constant (A001620).

A365172 The sum of divisors d of n such that gcd(d, n/d) is a square.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, 21, 18, 30, 20, 30, 32, 36, 24, 36, 26, 42, 28, 40, 30, 72, 32, 45, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 84, 50, 78, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 85, 84, 144, 68
Offset: 1

Views

Author

Amiram Eldar, Aug 25 2023

Keywords

Comments

The number of these divisors is A365171(n).

Links

Crossrefs

Cf. A000203, A005117, A034448, A035316, A046101, A365171, A365174.

Programs

  • Mathematica
    f[p_, e_] := If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), (1 + p^(2*Floor[(e + 1)/4] + 1))*(p^(2*Floor[e/4] + 2) - 1)/(p^2 - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(e%2, (1 + p^(2*((e+1)\4)+1))*(p^(2*(e\4)+2) - 1)/(p^2 - 1), (p^(e+2) - 1)/(p^2 - 1)));}

Formula

Multiplicative with a(p^e) = (p^(e+2) - 1)/(p^2 - 1) if e is even, and (1 + p^(2*floor((e+1)/4) + 1))*(p^(2*floor(e/4)+2) - 1)/(p^2 - 1) if e is odd.
a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).
a(n) >= A034448(n), with equality if and only if n is not a biquadrateful number (A046101).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1/(2 * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^5)) = 0.696082796052... .

A386470 The number of divisors of n whose exponents in their prime factorization are squares.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 3, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 6, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 3, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 4, 8, 2, 6, 3, 4, 2, 8, 4, 4, 4
Offset: 1

Views

Author

Amiram Eldar, Jul 22 2025

Keywords

Comments

First differs from A365171 and A369310 at n = 32.
First differs from A365488 at n = 128.
The number of terms in A197680 that divide n.
The sum of these divisors is A386471(n) and the largest of them is A386469(n).

Links

Crossrefs

Cf. A000005, A005117, A197680, A365171, A365488, A369310, A386469, A386471.

Programs

  • Mathematica
    f[p_, e_] := Floor[Sqrt[e]] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecprod(apply(x -> sqrtint(x) + 1, factor(n)[, 2]));

Formula

Multiplicative with a(p^e) = A048760(e) + 1.
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).

A377707 Numbers k that have a record number of divisors d such that gcd(d, k/d) is a square.

Original entry on oeis.org

1, 2, 6, 30, 210, 1680, 2310, 18480, 30030, 240240, 480480, 3843840, 4084080, 8168160, 65345280, 77597520, 155195040, 1241560320, 1784742960, 3569485920, 28555887360, 51757545840, 103515091680, 828120733440, 1604483921040, 3208967842080, 25671742736640, 51343485473280
Offset: 1

Views

Author

Amiram Eldar, Nov 04 2024

Keywords

Comments

Indices of records in A365171.
The corresponding record values are 1, 2, 4, 8, 16, 24, 32, 48, 64, 96, ... (see the link for more values).

Links

Crossrefs

Cf. A365171.
Subsequence of A025487.

Programs

  • Mathematica
    f[p_, e_] := Floor[(e + 3)/4] + Floor[(e + 4)/4]; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq
Showing 1-6 of 6 results.

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