A358340 a(n) is the smallest n-digit number whose fourth power is zeroless.

1, 11, 104, 1027, 10267, 102674, 1026708, 10266908, 102669076, 1026690113, 10266901031, 102669009704, 1026690096087, 10266900960914, 102669009608176, 1026690096080369, 10266900960803447, 102669009608034434, 1026690096080341627, 10266900960803409734, 102669009608034097731, 1026690096080340972491

Offset: 1

Author

Mohammed Yaseen, Nov 10 2022

Keywords

• nonn
• base

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

Crossrefs

Cf. A052382, A052040, A052044.

Cf. A253643, A252484, A253644, A253647, A124648, A124649.

Programs

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

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

Formula

a(n) ~ 10^(n + 1/4) / sqrt(3).

Extensions

More terms from David A. Corneth, Nov 10 2022