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

1,1

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.

Links

Felix Huber, Table of n, a(n) for n = 1..10000

Wikipedia, Significant Figures

Formula

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

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

Example

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.

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);

Prog

(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

Crossrefs

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

Sequence in context: A108271 A351121 A245585 * A343741 A060269 A395346

Adjacent sequences: A384920 A384921 A384922 * A384924 A384925 A384926

Keyword

nonn,base

Author

Felix Huber, Jun 26 2025

Status

approved