A084007 a(n) = A084006(n)^(1/2).

6, 9, 33, 66, 99, 333, 666, 999, 3333, 6666, 9999, 33333, 66666, 99999, 333333, 666666, 999999, 3333333, 6666666, 9999999, 33333333, 66666666, 99999999, 333333333, 444444444, 555555555, 666666666, 777777777, 888888888, 999999999

Offset: 0

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 23 2003

Keywords

• base
• nonn

Crossrefs

Cf. A084004, A084005, A084006.

Programs

Python

from itertools import count, islice
from math import prod, isqrt
from sympy import factorint
def A084007_gen(): # generator of terms
    for l in count(1):
        m = 10**l-1
        x = prod(p for p, e in factorint(m).items() if e&1)
        y = isqrt(x*m)
        yield from (j*y for j in range(isqrt(10**(l-1)//x)+1,isqrt(m//x)+1))
A084007_list = list(islice(A084007_gen(),30)) # Chai Wah Wu, Mar 20 2025

Formula

Pattern exhibited by early terms does not continue without interruption. First disruption occurs at a(25)=444444444. Terms with k-digits exhibit the earlier pattern where (10^k-1)/9 is squarefree and k=9 is the first occurrence where (10^k-1)/9 is not squarefree. Others occur at k=18, 22, 27, 36, 42, 44, 45. - Ray Chandler, Aug 04 2003

Extensions

More terms from Ray Chandler, May 31 2003

More terms from Ray Chandler, Aug 04 2003