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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.

%I A384923 #17 Jul 02 2025 03:26:32
%S A384923 19,23,37,39,45,36,27,17,25,15,36,19,20,36,25,37,28,13,27,52,39,17,38,
%T A384923 27,26,17,23,24,37,19,25,26,26,41,58,57,25,12,25,22,24,19,33,48,23,41,
%U A384923 49,23,32,32,23,30,19,17,31,27,24,47,24,26,18,22,19,48,31,22
%N 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.
%C A384923 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.
%H A384923 Felix Huber, <a href="/A384923/b384923.txt">Table of n, a(n) for n = 1..10000</a>
%H A384923 Wikipedia, <a href="https://en.wikipedia.org/wiki/Significant_figures">Significant Figures</a>
%F A384923 a(n) >= max(10, A384924(n)).
%F A384923 a(A113507(k) - floor(sqrt(A113507(k)))) = 10 for positive integers k.
%e A384923 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.
%p A384923 A384923:=proc(n)
%p A384923     local m,b,k;
%p A384923     m:=n+floor(1/2+sqrt(n));
%p A384923     b:=floor(log10(sqrt(m)));
%p A384923     k:=9-b;
%p A384923     while nops(convert(ListTools:-Reverse(convert(floor(10^k*sqrt(m)),'base',10)),set))<10 do
%p A384923         k:=k+1
%p A384923     od;
%p A384923     return k+b+1
%p A384923 end proc;
%p A384923 seq(A384923(n),n=1..66);
%o A384923 (Python)
%o A384923 from itertools import count
%o A384923 from math import isqrt
%o A384923 def A384923(n):
%o A384923     m = n+(k:=isqrt(n))+(n>k*(k+1))
%o A384923     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
%Y A384923 Cf. A000037, A000196, A003285, A061845, A113507, A384924.
%K A384923 nonn,base
%O A384923 1,1
%A A384923 _Felix Huber_, Jun 26 2025