OFFSET
1,6
COMMENTS
For each k >= 0 the sequence of solutions to a(x) = k has a positive asymptotic density (Shiu, 1991).
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
P. Shiu, The distribution of cube-full numbers, Glasgow Mathematical Journal, Vol. 33, No. 3 (1991), pp. 287-295. See section 3, p. 291.
FORMULA
Asymptotic mean: lim_{m->oo} (1/m) Sum_{k=1..m} a(k) = A362974 - 1 = 3.659266... . - Amiram Eldar, May 11 2023
EXAMPLE
a(2) = 1 since there is one cubefull number, 16 = 2^4, between 2^3 = 8 and 3^3 = 27.
MATHEMATICA
cubQ[n_] := Min[FactorInteger[n][[;; , 2]]] > 2; a[n_] := Count[Range[n^3 + 1, (n + 1)^3 - 1], _?cubQ]; Array[a, 100]
PROG
(Python)
from math import gcd
from sympy import integer_nthroot, factorint
def A337736(n):
def f(x):
c = 0
for w in range(1, integer_nthroot(x, 5)[0]+1):
if all(d<=1 for d in factorint(w).values()):
for y in range(1, integer_nthroot(z:=x//w**5, 4)[0]+1):
if gcd(w, y)==1 and all(d<=1 for d in factorint(y).values()):
c += integer_nthroot(z//y**4, 3)[0]
return c
return f((n+1)**3-1)-f(n**3) # Chai Wah Wu, Sep 10 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Sep 17 2020
STATUS
approved