A365346 -id:A365346 - cp's OEIS frontend

A365345 The number of divisors of the smallest square divisible by n.

1, 3, 3, 3, 3, 9, 3, 5, 3, 9, 3, 9, 3, 9, 9, 5, 3, 9, 3, 9, 9, 9, 3, 15, 3, 9, 5, 9, 3, 27, 3, 7, 9, 9, 9, 9, 3, 9, 9, 15, 3, 27, 3, 9, 9, 9, 3, 15, 3, 9, 9, 9, 3, 15, 9, 15, 9, 9, 3, 27, 3, 9, 9, 7, 9, 27, 3, 9, 9, 27, 3, 15, 3, 9, 9, 9, 9, 27, 3, 15, 5, 9, 3

Offset: 1

Amiram Eldar, Sep 02 2023

The sum of these divisors is A365346(n).

The number of divisors of the square root of the smallest square divisible by n is A322483(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000005, A053143, A322483, A365346, A365489, A365492.

f[p_, e_] := e + 1 + Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> x + 1 + x%2, factor(n)[, 2]));

a(n) = numdiv(n*core(n)); \\ Michel Marcus, Sep 02 2023

a(n) = A000005(A053143(n)).
Multiplicative with a(p^e) = e + 1 + (e mod 2).
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 + 2/p^s - 1/p^(2*s)).
From Vaclav Kotesovec, Sep 05 2023: (Start)
Dirichlet g.f.: zeta(s)^3 * zeta(2*s) * Product_{p prime} (1 - 4/p^(2*s) + 4/p^(3*s) - 1/p^(4*s)).
Let f(s) = Product_{primes p} (1 - 4/p^(2*s) + 4/p^(3*s) - 1/p^(4*s)).
Sum_{k=1..n} a(k) ~ f(1) * Pi^2 * n / 6 * (log(n)^2/2 + (3*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)) * log(n) + 1 - 3*gamma + 3*gamma^2 - 3*sg1 + (3*gamma - 1)*12*zeta'(2)/Pi^2 + 12*zeta''(2)/Pi^2 + (12*zeta'(2)/Pi^2 + 3*gamma - 1)*f'(1)/f(1) + f''(1)/(2*f(1))), where
f(1) = Product_{primes p} (1 - 4/p^2 + 4/p^3 - 1/p^4) = 0.2177787166195363783230075141194468131307977550013559376482764035236264911...,
f'(1) = f(1) * Sum_{primes p} 4*(2*p - 1) * log(p) / (1 - 3*p + p^2 + p^3) = 0.7343690473711153863995729489689746152413988981744946512300478410459132782...
f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} 4*p*(-1 + 2*p + p^2 - 4*p^3) * log(p)^2 / (1 - 3*p + p^2 + p^3)^2 = 0.1829055032494906699795154632343894745397324334876662084674149254022564139...,
gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)

A365347 The sum of divisors of the smallest number whose square is divisible by n.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 7, 4, 18, 12, 12, 14, 24, 24, 7, 18, 12, 20, 18, 32, 36, 24, 28, 6, 42, 13, 24, 30, 72, 32, 15, 48, 54, 48, 12, 38, 60, 56, 42, 42, 96, 44, 36, 24, 72, 48, 28, 8, 18, 72, 42, 54, 39, 72, 56, 80, 90, 60, 72, 62, 96, 32, 15, 84, 144, 68, 54

Offset: 1

Amiram Eldar, Sep 02 2023

The number of divisors of the smallest number whose square is divisible by n is A322483(n).

The sum of divisors of the smallest square divisible by n is A365346(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A019554, A322483, A365346.

Cf. A002117, A065465.

f[p_, e_] := (p^((e + Mod[e, 2])/2 + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^((f[i,2] + f[i,2]%2)/2 + 1) - 1)/(f[i,1] - 1));}

a(n) = sigma(n/core(n, 1)[2]); \\ Michel Marcus, Sep 02 2023

a(n) = A000203(A019554(n)).
Multiplicative with a(p^e) = (p^(e + 1 + (e mod 2)) - 1)/(p - 1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * zeta(3) * Product_{p prime} (1 - 1/(p^2*(p+1))) = (1/2) * A002117 * A065465 = 0.529814898136... .

A365479 The sum of unitary divisors of the smallest square divisible by n.

Original entry on oeis.org

1, 5, 10, 5, 26, 50, 50, 17, 10, 130, 122, 50, 170, 250, 260, 17, 290, 50, 362, 130, 500, 610, 530, 170, 26, 850, 82, 250, 842, 1300, 962, 65, 1220, 1450, 1300, 50, 1370, 1810, 1700, 442, 1682, 2500, 1850, 610, 260, 2650, 2210, 170, 50, 130, 2900, 850, 2810, 410

Offset: 1

Amiram Eldar, Sep 05 2023

The number of unitary divisors of the smallest square divisible by n is the same as the number of unitary divisors of n, A034444(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002117, A034444, A034448, A053143, A365346, A365480, A365481.

f[p_, e_] := 1 + p^(e + Mod[e, 2]); a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i=1, #f~, f[i,1]^(f[i,2] + f[i,2]%2) + 1);}

from math import prod
from sympy import factorint
def A365479(n): return prod(p**(e+(e&1))+1 for p,e in factorint(n).items()) # Chai Wah Wu, Sep 05 2023

a(n) = A034448(A053143(n)).
Multiplicative with a(p^e) = p^(e + (e mod 2)) + 1.
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-2) - 1/p^(3*s-2)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/45) * zeta(3) * Product_{p prime} (1 - 1/p^4 + 1/p^5 - 1/p^6) = 0.248414056414... .

cp's OEIS Frontend

A365345 The number of divisors of the smallest square divisible by n.

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A365347 The sum of divisors of the smallest number whose square is divisible by n.

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Author

Keywords

Comments

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Crossrefs

Programs

Mathematica

PARI

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A365479 The sum of unitary divisors of the smallest square divisible by n.

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Author

Keywords

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Programs

Mathematica

PARI

Python

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