A365479 -id:A365479 - cp's OEIS frontend

A365480 The sum of unitary divisors of the smallest exponentially odd number that is divisible by n.

1, 3, 4, 9, 6, 12, 8, 9, 28, 18, 12, 36, 14, 24, 24, 33, 18, 84, 20, 54, 32, 36, 24, 36, 126, 42, 28, 72, 30, 72, 32, 33, 48, 54, 48, 252, 38, 60, 56, 54, 42, 96, 44, 108, 168, 72, 48, 132, 344, 378, 72, 126, 54, 84, 72, 72, 80, 90, 60, 216, 62, 96, 224, 129, 84

Offset: 1

Amiram Eldar, Sep 05 2023

The number of unitary divisors of the smallest exponentially odd number that is 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
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Cf. A034444, A034448, A356191, A365349, A365479, A365481.

f[p_, e_] := p^(e + 1 - Mod[e, 2]) + 1; 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] + 1 - f[i,2]%2) + 1);}

a(n) = A034448(A356191(n)).
Multiplicative with a(p^e) = p^(e + 1 - (e mod 2)) + 1.
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-3) - 1/p^(2*s-2) - 1/p^(2*s-1) - 1/p^(3*s-3)).
From Vaclav Kotesovec, Sep 05 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(2*s-2) * zeta(2*s-3) * Product_{p prime} (1 - p^(7-6*s) - p^(5-5*s) + p^(7-5*s) + 2*p^(4-4*s) + 2*p^(5-4*s) - p^(6-4*s) + p^(2-3*s) - p^(4-3*s) - p^(1-2*s) - 2*p^(2-2*s)).
Let f(s) = Product_{p prime} (1 - p^(7-6*s) - p^(5-5*s) + p^(7-5*s) + 2*p^(4-4*s) + 2*p^(5-4*s) - p^(6-4*s) + p^(2-3*s) - p^(4-3*s) - p^(1-2*s) - 2*p^(2-2*s)).
Sum_{k=1..n} a(k) ~ n^2 * Pi^4 * f(2) / 144 * (log(n) + 3*gamma - 1/2 + 18*zeta'(2)/Pi^2 + f'(2)/f(2)), where
f(2) = Product_{p prime} (1 - 4/p^2 + 2/p^3 + 3/p^4 - 2/p^5) = 0.17432153313226756485612314112586411632220602294650993976966957787608316...,
f'(2) = f(2) * Sum_{p prime} 11 * log(p) / (p^2 + p - 2) = f(2) * 5.12969275236278527949034734003948649118572887258486718244613616120875581...
and gamma is the Euler-Mascheroni constant A001620. (End)

A365481 The sum of unitary divisors of the smallest number whose square is divisible by n.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 5, 4, 18, 12, 12, 14, 24, 24, 5, 18, 12, 20, 18, 32, 36, 24, 20, 6, 42, 10, 24, 30, 72, 32, 9, 48, 54, 48, 12, 38, 60, 56, 30, 42, 96, 44, 36, 24, 72, 48, 20, 8, 18, 72, 42, 54, 30, 72, 40, 80, 90, 60, 72, 62, 96, 32, 9, 84, 144, 68, 54, 96

Offset: 1

Amiram Eldar, Sep 05 2023

The number of unitary divisors of the smallest number whose square is 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, A013661, A019554, A034444, A034448, A365347, A365479, A365480.

f[p_, e_] := p^Ceiling[e/2] + 1; 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]^ceil(f[i,2]/2) + 1);}

from math import prod
from sympy import factorint
def A365481(n): return prod(p**((e>>1)+(e&1))+1 for p,e in factorint(n).items()) # Chai Wah Wu, Sep 05 2023

a(n) = A034448(A019554(n)).
Multiplicative with a(p^e) = p^(ceiling(e/2)) + 1.
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-1) - 1/p^(3*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * zeta(2) * zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4 - 1/p^5 + 1/p^6) = 0.515959523197... .

cp's OEIS Frontend

A365480 The sum of unitary divisors of the smallest exponentially odd number that is divisible by n.

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