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

%I A358340 #22 Feb 16 2025 08:34:04
%S A358340 1,11,104,1027,10267,102674,1026708,10266908,102669076,1026690113,
%T A358340 10266901031,102669009704,1026690096087,10266900960914,
%U A358340 102669009608176,1026690096080369,10266900960803447,102669009608034434,1026690096080341627,10266900960803409734,102669009608034097731,1026690096080340972491
%N A358340 a(n) is the smallest n-digit number whose fourth power is zeroless.
%C A358340 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.
%C A358340 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).
%H A358340 Michael S. Branicky, <a href="/A358340/b358340.txt">Table of n, a(n) for n = 1..69</a>
%H A358340 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Zerofree.html">Zerofree</a>
%F A358340 a(n) ~ 10^(n + 1/4) / sqrt(3).
%o A358340 (Python)
%o A358340 from itertools import count
%o A358340 from sympy import integer_nthroot
%o A358340 def a(n):
%o A358340     start = integer_nthroot(int("1"*(4*(n-1)+1)), 4)[0]
%o A358340     return next(i for i in count(start) if "0" not in str(i**4))
%o A358340 print([a(n) for n in range(1, 22)]) # _Michael S. Branicky_, Nov 10 2022
%o A358340 (PARI) a(n) = my(x=10^(n-1)); while(! vecmin(digits(x^4)), x++); x; \\ _Michel Marcus_, Nov 10 2022
%o A358340 (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
%Y A358340 Cf. A052382, A052040, A052044.
%Y A358340 Cf. A253643, A252484, A253644, A253647, A124648, A124649.
%K A358340 nonn,base
%O A358340 1,2
%A A358340 _Mohammed Yaseen_, Nov 10 2022
%E A358340 More terms from _David A. Corneth_, Nov 10 2022