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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A386469 The largest divisor of n whose exponents in its prime factorization are squares.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 16, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 16, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 48, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 16, 65, 66, 67, 34, 69, 70
Offset: 1

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Author

Amiram Eldar, Jul 22 2025

Keywords

Comments

The largest term in A197680 that divides n.
The number of these divisors is A386470(n) and their sum is A386471(n).

Links

Crossrefs

Cf. A048760, A197680, A386470, A386471.
Similar sequences: A008833, A350390, A365683.

Programs

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

Formula

Multiplicative with a(p^e) = p^A048760(e).
a(n) <= n, with equality if and only if n is in A197680.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} Sum_{k>=2} (1/p^(k^2-1) - 1/p^(k^2-2)) = 0.74491327356409794092... .

A366765 The largest divisor of n that have no exponent 2 in their prime factorization.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 6, 13, 14, 15, 16, 17, 6, 19, 10, 21, 22, 23, 24, 5, 26, 27, 14, 29, 30, 31, 32, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 22, 15, 46, 47, 48, 7, 10, 51, 26, 53, 54, 55, 56, 57, 58, 59, 30, 61, 62, 21, 64, 65, 66, 67, 34, 69
Offset: 1

Views

Author

Amiram Eldar, Oct 21 2023

Keywords

Comments

The largest term of A337050 that divides n.
The number of these divisors is A366763(n), and their sum is A366764(n).

Links

Crossrefs

Cf. A337050, A366763, A366764.
Similar sequences: A055231, A057521, A008833, A350390.

Programs

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

Formula

Multiplicative with a(p^e) = p if e <= 2 and p^e otherwise.
a(n) <= n, with equality if and only if n is in A337050.
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^(2*s-2) + 1/p^(2*s-1) + 1/p^(3*s-3) - 1/p^(3*s-2)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 2/p^3 - 1/p^4) = 0.83234421330425224469... .

A368170 The largest cubefull exponentially odd divisor of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 27, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 14 2023

Keywords

Comments

First differs from A008834 at n = 32, and from A366906 at n = 64.

Links

Crossrefs

Cf. A004709, A335988, A350390, A356192, A368167, A368171.
Cf. A008834, A366906.

Programs

  • Mathematica
    f[p_, e_] := If[e <= 2, 1, If[EvenQ[e], p^(e-1), p^e]]; 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] <= 2, 1, if(!(f[i,2]%2), f[i,1]^(f[i, 2]-1), f[i, 1]^f[i, 2])))};

Formula

Multiplicative with a(p^e) = 1 if e <= 2, a(p^e) = p^e if e is odd and e > 1, and p^(e-1) otherwise.
a(n) = n/A368171(n).
a(n) >= 1, with equality if and only if n is cubefree (A004709).
a(n) <= n, with equality if and only if n is in A335988.

A349326 a(n) is the number of prime powers (not including 1) that are bi-unitary divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 3, 2, 1, 3, 1, 5, 2, 2, 2, 2, 1, 2, 2, 4, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 2, 4, 2, 2, 1, 3, 1, 2, 2, 5, 2, 3, 1, 2, 2, 3, 1, 4, 1, 2, 2, 2, 2, 3, 1, 4, 3, 2, 1, 3, 2, 2, 2, 4, 1, 3, 2, 2, 2, 2, 2, 6, 1, 2, 2, 2, 1, 3, 1, 4, 3
Offset: 1

Views

Author

Amiram Eldar, Nov 15 2021

Keywords

Comments

The total number of prime powers (not including 1) that divide n is A001222(n).
The least number k such that a(k) = m is A122756(m).

Examples

			12 has 4 bi-unitary divisors, 1, 3, 4 and 12. Two of these divisors, 3 and 4 = 2^2 are prime powers. Therefore a(12) = 2.

Links

Crossrefs

Cf. A000961, A001222, A122756, A162641, A222266, A246655, A286324, A268335, A350390.
Similar sequences: A001222, A349258, A349281.
Cf. A083342, A179119.

Programs

  • Mathematica
    f[p_, e_] := If[OddQ[e], e, e - 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecsum(apply(x -> if(x%2, x, x-1), factor(n)[, 2])); \\ Amiram Eldar, Sep 29 2023

Formula

Additive with a(p^e) = e if e is odd, and e-1 if e is even.
a(n) <= A001222(n), with equality if and only if n is an exponentially odd number (A268335).
a(n) <= A286324(n) - 1, with equality if and only if n is a prime power (including 1, A000961).
a(n) = A001222(n) - A162641(n). - Amiram Eldar, May 18 2023
From Amiram Eldar, Sep 29 2023: (Start)
a(n) = A001222(A350390(n)) (the number of prime factors of the largest exponentially odd number dividing n, counted with multiplicity).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B_2 - C), where B_2 = A083342 and C = A179119. (End)

A365337 The sum of divisors of the largest exponentially odd number dividing n.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 15, 4, 18, 12, 12, 14, 24, 24, 15, 18, 12, 20, 18, 32, 36, 24, 60, 6, 42, 40, 24, 30, 72, 32, 63, 48, 54, 48, 12, 38, 60, 56, 90, 42, 96, 44, 36, 24, 72, 48, 60, 8, 18, 72, 42, 54, 120, 72, 120, 80, 90, 60, 72, 62, 96, 32, 63, 84, 144, 68
Offset: 1

Views

Author

Amiram Eldar, Sep 01 2023

Keywords

Comments

The number of divisors of the largest exponentially odd number dividing n is A286324(n).

Links

Crossrefs

Cf. A000203, A065487, A286324, A350390.

Programs

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

Formula

a(n) = A000203(A350390(n)).
Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e is odd and (p^e-1)/(p-1) if e is even.
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-2) + 1/p^(3*s-2)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p*(p^2-1))) = 1.2312911488886... (A065487). - Amiram Eldar, Sep 01 2023
Previous Showing 11-15 of 15 results.

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