OFFSET
1,2
COMMENTS
It has been proved that there exist infinitely many zeroless squares and cubes but there is apparently no proof for 4th powers, 5th powers, etc.
This sequence approaches the decimal expansion of 9000^(-1/4). Similar sequences of other small powers k seem to approach the decimal expansion of (9*10^(k-1))^(-1/k).
LINKS
Michael S. Branicky, Table of n, a(n) for n = 1..69
Eric Weisstein's World of Mathematics, Zerofree
FORMULA
a(n) ~ 10^(n + 1/4) / sqrt(3).
PROG
(Python)
from itertools import count
from sympy import integer_nthroot
def a(n):
start = integer_nthroot(int("1"*(4*(n-1)+1)), 4)[0]
return next(i for i in count(start) if "0" not in str(i**4))
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Nov 10 2022
(PARI) a(n) = my(x=10^(n-1)); while(! vecmin(digits(x^4)), x++); x; \\ Michel Marcus, Nov 10 2022
(PARI) a(n) = { my(s = sqrtnint(10^(4*n - 3) \ 9, 4)); for(i = s, oo, c = i^4; if(vecmin(digits(c)) > 0, return(i) ) ) } \\ David A. Corneth, Nov 10 2022
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Mohammed Yaseen, Nov 10 2022
EXTENSIONS
More terms from David A. Corneth, Nov 10 2022
STATUS
approved