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
KEYWORD
nonn
AUTHOR
Michel Marcus, Jul 21 2013
STATUS
approved