A384923 a(n) is the smallest number of leading significant digits of the square root of the n-th nonsquare that includes all decimal digits.

19, 23, 37, 39, 45, 36, 27, 17, 25, 15, 36, 19, 20, 36, 25, 37, 28, 13, 27, 52, 39, 17, 38, 27, 26, 17, 23, 24, 37, 19, 25, 26, 26, 41, 58, 57, 25, 12, 25, 22, 24, 19, 33, 48, 23, 41, 49, 23, 32, 32, 23, 30, 19, 17, 31, 27, 24, 47, 24, 26, 18, 22, 19, 48, 31, 22

Offset: 1

Author

Felix Huber, Jun 26 2025

Keywords

• nonn
• base

Comments

Squares are excluded by definition because a(n) would only exist for positive integers s that include all decimal digits. The smallest square s^2 for which a(n) would exist is 1023456789^2 = 1047463798950190521.

Examples

			The leading 19 significant digits of sqrt(2) are [1, 4, 1, 4, 2, 1, 3, 5, 6, 2, 3, 7, 3, 0, 9, 5, 0, 4, 8]. These digits include all decimal digits, with the digit '8' appearing for the first time at position 19. Since 2 is the first nonsquare, it follows that a(1) = 19.
		

Links

• Felix Huber, Table of n, a(n) for n = 1..10000
• Wikipedia, Significant Figures

Crossrefs

Cf. A000037, A000196, A003285, A061845, A113507, A384924.

Programs

Maple

A384923:=proc(n)
    local m,b,k;
    m:=n+floor(1/2+sqrt(n));
    b:=floor(log10(sqrt(m)));
    k:=9-b;
    while nops(convert(ListTools:-Reverse(convert(floor(10^k*sqrt(m)),'base',10)),set))<10 do
        k:=k+1
    od;
    return k+b+1
end proc;
seq(A384923(n),n=1..66);

Python

from itertools import count
from math import isqrt
def A384923(n):
    m = n+(k:=isqrt(n))+(n>k*(k+1))
    return 1+next(n for n in count(9) if len(set(str(isqrt(10**(n<<1)*m))))==10) # Chai Wah Wu, Jul 01 2025

Formula

a(n) >= max(10, A384924(n)).

a(A113507(k) - floor(sqrt(A113507(k)))) = 10 for positive integers k.