A380396 a(n) is the sum of the unitary divisors of n that are cubes.

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 65, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Jan 23 2025

Keywords

• nonn
• easy
• mult

Comments

The number of unitary divisors of n that are cubes is A380395(n).

Examples

			a(8) = 9 since 8 has 2 unitary divisors that are cubes, 1 = 1^3 and 8 = 2^3, and 1 + 8 = 9.
a(216) = 252 since 216 has 4 unitary divisors that are cubes, 1 = 1^3, 8 = 2^3, 27 = 3^3 and 216 = 6^3, and 1 + 8 + 27 + 216 = 252.
		

Links

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

Crossrefs

Cf. A000578, A046100, A077610, A113061, A358347, A366761, A371334, A380395.

Programs

Mathematica

f[p_, e_] := If[Divisible[e, 3], p^e + 1, 1]; 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]%3, 1, f[i, 1]^f[i, 2] + 1));}

Formula

a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * [d is cube], where [] is the Iverson bracket.

a(n) >= 1, with equality if and only if n is not in A366761.

a(n) <= A113061(n), with equality if and only if n is biquadratefree (A046100).

Multiplicative with a(p^e) = p^e + 1 if e is divisible by 3, and 1 otherwise.

Sum_{k=1..n} a(k) ~ c * n^(4/3) / 4, where c = zeta(4/3)/zeta(7/3) = 2.54455250463133711749... .

Dirichlet g.f.: zeta(s) * zeta(3*s-3) / zeta(4*s-3).

In general, the average order of the sum of the unitary divisors that are m-powers is c * n^(1+1/m) / (m+1), where c = zeta(1+1/m)/zeta(2+1/m), and its Dirichlet g.f. is zeta(s) * zeta(m*s-m) / zeta((m+1)*s-m), both for m >= 2.