|
|
A160112
|
|
Number of cubefree integers not exceeding 10^n.
|
|
5
|
|
|
1, 9, 85, 833, 8319, 83190, 831910, 8319081, 83190727, 831907372, 8319073719, 83190737244, 831907372522, 8319073725828, 83190737258105, 831907372580692, 8319073725807178, 83190737258070643, 831907372580707771
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
An alternate definition specifying "less than 10^n" would yield the same sequence except for the first 3 terms: 0, 8, 84, 833, 8319, etc. (since powers of 10 beyond 1000 are not cubefree anyhow).
The limit of a(n)/10^n is the inverse of Apery's constant, 1/zeta(3), whose digits are given by A088453.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = Sum_{i=1..floor(10^(n/3))} A008683(i)*floor(10^n/i^3).
|
|
EXAMPLE
|
a(0)=1 because 1 <= 10^0 is not a multiple of the cube of a prime.
a(1)=9 because the 9 numbers 1,2,3,4,5,6,7,9,10 are cubefree; 8 is not.
a(2)=85 because there are 85 cubefree integers equal to 100 or less.
a(3)=833 because there are 833 cubefree integers below 1000 (which is not cubefree itself).
|
|
MAPLE
|
|
|
MATHEMATICA
|
Table[ Sum[ MoebiusMu[x]*Floor[10^n/(x^3)], {x, 10^(n/3)}], {n, 0, 18}] (* Robert G. Wilson v, May 27 2009 *)
|
|
PROG
|
(Python)
from sympy import mobius, integer_nthroot
def A160112(n): return sum(mobius(k)*(10**n//k**3) for k in range(1, integer_nthroot(10**n, 3)[0]+1)) # Chai Wah Wu, Aug 06 2024
(Python)
from bitarray import bitarray
from sympy import integer_nthroot
q = 10**n
m = integer_nthroot(q, 3)[0]+1
a, b = bitarray(m), bitarray(m)
a[1], p, i, c = 1, 2, 4, q-sum(q//k**3 for k in range(2, m))
while i < m:
j = 2
while i < m:
if j==p:
c -= (b[i]^1 if a[i] else -1)*(q//i**3)
j, a[i], b[i] = 0, 1, 1
else:
t1, t2 = a[i], b[i]
if (t1&t2)^1:
a[i], b[i] = (t1^1)&t2, ((t1^1)&t2)^1
c += (t2 if t1 else 2)*(q//i**3) if (t1^1)&t2 else (t2-2 if t1 else 0)*(q//i**3)
i += p
j += 1
p += 1
while a[p]|b[p]:
p += 1
i = p<<1
|
|
CROSSREFS
|
A004709 (cubefree numbers). A088453 (limit of the string of digits). A160113 (binary counterpart for cubefree integers). A071172 & A053462 (decimal counterpart for squarefree integers). A143658 (binary counterpart for squarefree integers).
|
|
KEYWORD
|
easy,nice,nonn,changed
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|