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.

Previous Showing 21-25 of 25 results.

A368247 The number of cubefree divisors of the cubefull part of n (A360540).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 19 2023

Keywords

Links

Crossrefs

Cf. A004709, A036966, A073184, A323308, A360540.

Programs

  • Mathematica
    f[p_, e_] := If[e > 2, 3, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecprod(apply(x -> if(x < 3, 1, 3), factor(n)[, 2]));

Formula

a(n) = A073184(A360540(n)).
Multiplicative with a(p^e) = 1 if e <= 2, and 3 otherwise.
a(n) >= 1, with equality if and only if n is cubefree (A004709).
a(n) <= A073184(n), with equality if and only if n is cubefull (A036966).
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 2/p^(3*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 2/p^3) = 1.37700168952903630206... .
In general, the asymptotic mean of the number of k-free divisors of the k-full part of n is Product_{p prime} (1 + (k-1)/p^k).

A370239 The sum of divisors of n that are squares of squarefree numbers.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 26, 1, 10, 5, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 50, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 5, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 26, 5, 1, 1, 1, 5, 10, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Feb 13 2024

Keywords

Comments

The number of these divisors is A323308(n).

Links

Crossrefs

Cf. A005117, A036785, A048250, A062503, A071327, A078434, A323308, A370240.

Programs

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

Formula

Multiplicative with a(p) = 1 and a(p^e) = 1 + p^2 for e >= 2.
a(n) >= 1, with equality if and only if n is squarefree (A005117).
a(n) = A071327(n) + 1 if and only if n is not in A036785.
Dirichlet g.f.: zeta(s)*zeta(2*s-2)/zeta(4*s-4).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = 2*zeta(3/2)/Pi^2 = 0.5293779248... .

A381613 If n = Product (p_j^k_j) then a(n) = Product (min(p_j, k_j)), with a(1) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1
Offset: 1

Views

Author

Paolo Xausa, Mar 01 2025

Keywords

Comments

First differs from A323308 at n = 27.

Examples

			a(18) = 2 because 18 = 2^1*3^2, min(2,1) = 1, min(3,2) = 2 and 1*2 = 2.
a(300) = 4 because 300 = 2^2*3^1*5^2, min(2,2) = 2, min(3,1) = 1, min(5,2) = 2 and 2*1*2 = 4.

Links

Crossrefs

Cf. A008473, A008477, A035306, A323308, A369008, A381588, A381614.

Programs

  • Mathematica
    A381613[n_] := Times @@ Min @@@ FactorInteger[n];
Array[A381613, 100]
  • PARI
    a(n) = my(f=factor(n)); prod(i=1, #f~, min(f[i,1], f[i,2])); \\ Michel Marcus, Mar 02 2025

Formula

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + (1/p - 1/p^p)/(p-1)) = 1.59383299054679951264... . - Amiram Eldar, Mar 07 2025

A345371 Number of squarefree divisors of n whose square does not divide n.

Original entry on oeis.org

0, 1, 1, 0, 1, 3, 1, 0, 0, 3, 1, 2, 1, 3, 3, 0, 1, 2, 1, 2, 3, 3, 1, 2, 0, 3, 0, 2, 1, 7, 1, 0, 3, 3, 3, 0, 1, 3, 3, 2, 1, 7, 1, 2, 2, 3, 1, 2, 0, 2, 3, 2, 1, 2, 3, 2, 3, 3, 1, 6, 1, 3, 2, 0, 3, 7, 1, 2, 3, 7, 1, 0, 1, 3, 2, 2, 3, 7, 1, 2, 0, 3, 1, 6, 3, 3, 3, 2, 1, 6, 3, 2, 3
Offset: 1

Views

Author

Wesley Ivan Hurt, Jun 16 2021

Keywords

Examples

			a(30) = Sum_{d|30} mu(d)^2 * c(30/d^2) = 1*0 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 = 7.

Links

Crossrefs

Cf. A008683 (mu), A034444, A323308, A344137.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 1 &, SquareFreeQ[#] && ! Divisible[n, #^2] &]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)
  • PARI
    a(n) = {my(e = factor(n)[, 2]); 1 << #e - vecprod(apply(x -> min(x, 2), e));} \\ Amiram Eldar, Oct 13 2023

Formula

a(n) = Sum_{d|n} mu(d)^2 * c(n/d^2), where c(n) = ceiling(n) - floor(n).
a(n) = A034444(n) - A323308(n). - Amiram Eldar, Oct 13 2023

A370296 Inverse Moebius transform of A322327.

Original entry on oeis.org

1, 3, 3, 7, 3, 9, 3, 13, 7, 9, 3, 21, 3, 9, 9, 21, 3, 21, 3, 21, 9, 9, 3, 39, 7, 9, 13, 21, 3, 27, 3, 31, 9, 9, 9, 49, 3, 9, 9, 39, 3, 27, 3, 21, 21, 9, 3, 63, 7, 21, 9, 21, 3, 39, 9, 39, 9, 9, 3, 63, 3, 9, 21, 43, 9, 27, 3, 21, 9, 27, 3, 91, 3, 9, 21, 21, 9, 27, 3, 63, 21, 9, 3, 63
Offset: 1

Views

Author

Werner Schulte, Feb 14 2024

Keywords

Crossrefs

Cf. A322327, A000005, A323308.

Programs

  • Mathematica
    f[p_, e_] := e^2 + e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 14 2024 *)
  • PARI
    a(n) = factorback(apply(e->1+e+e^2,factor(n)[,2]))

Formula

Multiplicative with a(p^e) = 1 + e + e^2 for prime p and e >= 0.
Dirichlet g.f.: (zeta(s))^3 * zeta(2*s) / zeta(4*s).
Dirichlet inverse sequence b(n) for n > 0 is multiplicative with b(p) = -3 and b(p^e) = 2 * (-1)^((e+1)*(e+2)/2) for prime p and e > 1.
Dirichlet convolution of A000005 and A323308.
Previous Showing 21-25 of 25 results.

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

Content is available under The OEIS End-User License Agreement.