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 1-3 of 3 results.

A322483 The number of semi-unitary divisors of n.

Original entry on oeis.org

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

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Author

Amiram Eldar, Dec 11 2018

Keywords

Comments

The notion of semi-unitary divisor was introduced by Chidambaraswamy in 1967.
A semi-unitary divisor of n is defined as the largest divisor d of n such that the largest divisor of d that is a unitary divisor of n/d is 1. In terms of the relation defined in A322482, d is the largest divisor of n such that T(d, n/d) = 1 (the largest divisor d that is semiprime to n/d).
The number of divisors of n that are exponentially odd numbers (A268335). - Amiram Eldar, Sep 08 2023

Examples

			The semi-unitary divisors of 8 are 1, 2, 8 (4 is not semi-unitary divisor since the largest divisor of 4 that is a unitary divisor of 8/4 = 2 is 2 > 1), and their number is 3, thus a(8) = 3.

References

  • J. Chidambaraswamy, Sum functions of unitary and semi-unitary divisors, J. Indian Math. Soc., Vol. 31 (1967), pp. 117-126.

Links

Crossrefs

Cf. A000005, A005117, A019554, A034444, A268335, A286324, A322482.
Cf. A001620, A056671, A343443, A365488, A365491.

Programs

  • Mathematica
    f[p_, e_] := Floor[(e+3)/2]; sud[n_] := If[n==1, 1, Times @@ (f @@@ FactorInteger[n])]; Array[sud, 100]
  • PARI
    a(n) = {my(f = factor(n)); for (k=1, #f~, f[k,1] = (f[k,2]+3)\2; f[k,2] = 1;); factorback(f);} \\ Michel Marcus, Dec 14 2018
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * 1/(1-X^2) * (1 + X - X^2))[n], ", ")) \\ Vaclav Kotesovec, Sep 06 2023

Formula

Multiplicative with a(p^e) = floor((e+3)/2).
a(n) <= A000005(n) with equality if and only if n is squarefree (A005117).
a(n) = Sum_{d|n} mu(d/gcd(d, n/d))^2. - Ilya Gutkovskiy, Feb 21 2020
a(n) = A000005(A019554(n)) (the number of divisors of the smallest number whose square is divisible by n). - Amiram Eldar, Sep 02 2023
From Vaclav Kotesovec, Sep 06 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s)).
Dirichlet g.f.: zeta(s)^2 * zeta(2*s) * Product_{p prime} (1 - 2/p^(2*s) + 1/p^(3*s)).
Let f(s) = Product_{p prime} (1 - 2/p^(2*s) + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ Pi^2 * f(1) * n / 6 * (log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 2/p^2 + 1/p^3) = A065464 = 0.42824950567709444...,
f'(1) = f(1) * Sum_{p prime} (4*p-3) * log(p) / (p^3 - 2*p + 1) = 0.808661108949590913395... and gamma is the Euler-Mascheroni constant A001620. (End)

A365499 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s - 1/p^(4*s)).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 1, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 1, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 2, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 1, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 4, 8, 2, 2, 1, 4, 2, 8, 4, 4, 4
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 06 2023

Keywords

Comments

The number of unitary divisors of n that are biquadratefree numbers (A046100). - Amiram Eldar, Sep 06 2023

Links

Crossrefs

Cf. A001620, A046100, A056671, A365491, A365498.

Programs

  • Mathematica
    f[p_, e_] := If[e <= 3, 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 06 2023 *)
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * (1 + X - X^4))[n], ", "))

Formula

Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^(2*s) - 1/p^(4*s) + 1/p^(5*s)).
Let f(s) = Product_{p prime} (1 - 1/p^(2*s) - 1/p^(4*s) + 1/p^(5*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^4 + 1/p^5) = 0.576152735385667059520611078264117275406247116802896188543250284595724...,
f'(1) = f(1) * Sum_{p prime} (-5 + 4*p + 2*p^3) * log(p) / (1 - p - p^3 + p^5) = f(1) * 1.30114343965598023783147826007476613992233856698399986804189962...
and gamma is the Euler-Mascheroni constant A001620.
Multiplicative with a(p^e) = 2 if e <= 3, and 1 otherwise. - Amiram Eldar, Sep 06 2023

A365492 The number of divisors of the smallest 4th power divisible by n.

Original entry on oeis.org

1, 5, 5, 5, 5, 25, 5, 5, 5, 25, 5, 25, 5, 25, 25, 5, 5, 25, 5, 25, 25, 25, 5, 25, 5, 25, 5, 25, 5, 125, 5, 9, 25, 25, 25, 25, 5, 25, 25, 25, 5, 125, 5, 25, 25, 25, 5, 25, 5, 25, 25, 25, 5, 25, 25, 25, 25, 25, 5, 125, 5, 25, 25, 9, 25, 125, 5, 25, 25, 125, 5, 25
Offset: 1

Views

Author

Amiram Eldar, Sep 05 2023

Keywords

Comments

First differs from A082476 at n = 32.
The number of divisors of the 4th root of the smallest 4th power divisible by n, A053166(n), is A365491(n).

Links

Crossrefs

Cf. A000005, A053166, A053167, A082476, A365491.

Programs

  • Mathematica
    f[p_, e_] := 4*Ceiling[e/4] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecprod(apply(x -> 4*((x-1)\4) + 5, factor(n)[, 2]));

Formula

a(n) = A000005(A053167(n)).
Multiplicative with a(p^e) = 4*ceiling(e/4) + 1.
Dirichlet g.f.: zeta(s) * zeta(4*s) * Product_{p prime} (1 + 4/p^s - 1/p^(4*s)).
Showing 1-3 of 3 results.

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