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

Author

Amiram Eldar, Sep 05 2023

Keywords

• nonn
• easy
• mult

Comments

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).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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)