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

1,8

Comments

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

Links

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

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.

Example

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.

Mathematica

f[p_, e_] := If[Divisible[e, 3], p^e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%3, 1, f[i, 1]^f[i, 2] + 1)); }

Crossrefs

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

Sequence in context: A266557 A010534 A078297 * A189788 A264981 A370240

Adjacent sequences: A380393 A380394 A380395 * A380397 A380398 A380399

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 23 2025

Status

approved