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

1,2

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

Cf. A052382, A052040, A052044.

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

Sequence in context: A077250 A356128 A173851 * A295840 A158470 A163933

Adjacent sequences: A358337 A358338 A358339 * A358341 A358342 A358343

Keyword

nonn,base

Author

Mohammed Yaseen, Nov 10 2022

Extensions

More terms from David A. Corneth, Nov 10 2022

Status

approved