A368883 The number of infinitary divisors of n that are cubefree.

1, 2, 2, 2, 2, 4, 2, 3, 2, 4, 2, 4, 2, 4, 4, 1, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 3, 4, 2, 8, 2, 2, 4, 4, 4, 4, 2, 4, 4, 6, 2, 8, 2, 4, 4, 4, 2, 2, 2, 4, 4, 4, 2, 6, 4, 6, 4, 4, 2, 8, 2, 4, 4, 2, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 4, 4, 4, 8, 2, 2, 1, 4, 2, 8, 4, 4, 4

Offset: 1

Author

Amiram Eldar, Jan 09 2024

Keywords

• nonn
• easy
• mult

Comments

The number of infinitary divisors of n that are squarefree is A055076(n).

Links

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

Crossrefs

Cf. A000583, A004709, A037445, A055076, A077609.

Programs

Mathematica

f[p_, e_] := Switch[Mod[e, 4], 1, 2, 2, 2, 3, 3, 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecprod(apply(x -> [1,2,2,3][x%4+1], factor(n)[, 2]));

Formula

Multiplicative with a(p^e) = 2 if e == 1 or 2 (mod 4), 3 if e == 3 (mod 4), and 1 if e == 0 (mod 4).

a(n) >= 1, with equality if and only if n is a 4th power (A000583).

a(n) <= A037445(n), with equality if and only if n is cubefree (A004709).

Dirichlet g.f.: zeta(4*s) * Product_{p prime} (1 + 2/p^s + 2/p^(2*s) + 3/p^(3*s)).

From Vaclav Kotesovec, Jan 09 2024: (Start)

Dirichlet g.f.: zeta(4*s) * zeta(s)^2 * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s) - 4/p^(4*s) + 3/p^(5*s)).

Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s) - 4/p^(4*s) + 3/p^(5*s)).

Sum_{k=1..n} a(k) ~ f(1) * zeta(4) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1) + 4*zeta'(4)/zeta(4)), where

f(1) = Product_{p prime} (1 - 1/p^2 + 1/p^3 - 4/p^4 + 3/p^5) = 0.5857770602270641007515615375942370402509903724261557972367075945186871...,

f'(1) = f(1) * Sum_{p prime} (2*p^2 - p + 15) * log(p) / (p^4 + p^3 + p - 3) = f(1) * 1.319786264712492218167871116508220489817987315752197198819256094...,

gamma is the Euler-Mascheroni constant A001620, zeta(4) = Pi^4/90 = A013662 and for zeta'(4) see A261506. (End)