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%I #12 May 06 2022 13:13:51
%S 9,8,1,9,6,1,9,8,0,1,9,9,8,5,6,9,9,8,0,0,1,9,9,9,8,2,4,4,9,9,9,8,0,0,
%T 0,1,9,9,9,9,5,0,8,8,4,9,9,9,9,8,0,0,0,0,1,9,9,9,9,9,5,1,5,5,2,9,9,9,
%U 9,9,9,8,0,0,0,0,0,1,9,9,9,9,9,9,5,8
%N Constant whose decimal expansion is the concatenation of the largest n-digit square A061433(n), for n = 1, 2, 3, ...
%C The terms of sequence A339978 converge to this sequence of digits, and to this constant, up to powers of 10.
%H Harvey P. Dale, <a href="/A340207/b340207.txt">Table of n, a(n) for n = 0..1000</a>
%F c = 0.9819619801998569980019998244999800019999508849999800001999995155...
%F = Sum_{k >= 1} 10^(-k(k+1)/2)*floor(10^(k/2)-1)^2
%F a(-n(n+1)/2) = 9 for all n >= 2.
%e The largest square with 1, 2, 3, 4, ... digits is, respectively, 9 = 3^2, 81 = 9^2, 961 = 31^2, 9801 = 99^2, ....
%e Here we list the sequence of digits of these numbers: 9; 8, 1; 9, 6, 1; 9, 8, 0, 1; 9, 9, 8, 5, 6; ...
%e This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.98196198...
%t lnds[k_]:=Module[{c=Sqrt[10^k]},If[IntegerQ[c],(c-1)^2,Floor[c]^2]]; Flatten[IntegerDigits/@(lnds/@Range[15])] (* _Harvey P. Dale_, Dec 16 2021 *)
%o (PARI) concat([digits(sqrtint(10^k-1)^2)|k<-[1..14]]) \\ as seq. of digits
%o c(N=20)=sum(k=1,N,.1^(k*(k+1)/2)*sqrtint(10^k-1)^2) \\ as constant
%Y Cf. A061433 (largest n-digit square), A339978 (has this as "limit"), A340208 (same with "smallest n-digit cube", limit of A215692), A340209 (same for cubes, limit of A340115), A340220 (same for primes), A340219 (similar, with smallest primes, limit of A215641), A340222 (same for semiprimes), A340221 (same for smallest semiprimes, limit of A215647).
%Y Cf. A033307 (Champernowne constant), A030190 (binary), A001191 (concatenation of all squares), A134724 (cubes), A033308 (primes: Copeland-Erdős constant).
%K nonn,base,cons
%O 0,1
%A _M. F. Hasler_, Jan 01 2021
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