A224835 Sum of the cubes of the number of divisors function for those divisors of n that are less than or equal to the cube root of n.

1, 1, 1, 1, 1, 1, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 9, 9, 1, 17, 1, 9, 9, 9, 1, 17, 1, 9, 9, 9, 1, 17, 1, 9, 9, 9, 1, 17, 1, 9, 9, 9, 1, 17, 1, 9, 9, 9, 1, 17, 1, 9, 9, 36, 1, 17, 1, 36, 9, 9, 1, 44, 1, 9, 9

Offset

1,8

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Sary Drappeau, Propriétés multiplicatives des entiers friables translatés, arXiv:1307.4250 [math.NT] (see page 9).

B. Landreau, A New Proof of a Theorem of Van Der Corput, Bull. London Math. Soc. (1989) 21 (4): 366-368. doi: 10.1112/blms/21.4.366, see Lemma (3) page 1.

Formula

a(n) = (Sum_{d|n} d <= n^(1/3)) tau(d)^3.

If p is prime, a(p^k) = A000537(1 + floor(k/3)). - Robert Israel, Nov 30 2016

Example

a(7) = 1 because the divisors of 7 are 1 and 7; only 1 is less than the cube root of 7, and tau(1^3) = 1, so the sum is 1.
a(8) = 9 because the divisors of 8 are 1, 2, 4, 8; the cube root of 8 is 2, so only 1 and 2 are divisors less than or equal to the cube root, these divisors cubed are 1 and 8, which add up to 9.

Maple

f:= proc(n) add(numtheory:-tau(d)^3, d = select(t -> (t^3<=n), numtheory:-divisors(n))) end proc:
map(f, [$1..100]); # Robert Israel, Nov 30 2016

Mathematica

Table[selDivs = Select[Range[Floor[n^(1/3)]], IntegerQ[n/#]&]; Sum[DivisorSigma[0, selDivs[[m]]]^3, {m, Length[selDivs]}], {n, 100}] (* Alonso del Arte, Jul 21 2013 *)

Prog

(PARI) a(n) = sumdiv(n, d, (d^3<=n)*numdiv(d)^3) \\ Michel Marcus, Jul 21 2013

Crossrefs

Cf. A000537, A007425, A224834.

Sequence in context: A203130 A209050 A010690 * A329725 A339354 A280375

Adjacent sequences: A224832 A224833 A224834 * A224836 A224837 A224838

Keyword

nonn

Author

Michel Marcus, Jul 21 2013

Status

approved