A340207 Constant whose decimal expansion is the concatenation of the largest n-digit square A061433(n), for n = 1, 2, 3, ...

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, 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, 9, 9, 9, 8, 0, 0, 0, 0, 0, 1, 9, 9, 9, 9, 9, 9, 5, 8

Offset

0,1

Comments

The terms of sequence A339978 converge to this sequence of digits, and to this constant, up to powers of 10.

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Formula

c = 0.9819619801998569980019998244999800019999508849999800001999995155...

= Sum_{k >= 1} 10^(-k(k+1)/2)*floor(10^(k/2)-1)^2

a(-n(n+1)/2) = 9 for all n >= 2.

Example

The largest square with 1, 2, 3, 4, ... digits is, respectively, 9 = 3^2, 81 = 9^2, 961 = 31^2, 9801 = 99^2, ....
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; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.98196198...

Mathematica

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 *)

Prog

(PARI) concat([digits(sqrtint(10^k-1)^2)|k<-[1..14]]) \\ as seq. of digits
c(N=20)=sum(k=1, N, .1^(k*(k+1)/2)*sqrtint(10^k-1)^2) \\ as constant

Crossrefs

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).

Cf. A033307 (Champernowne constant), A030190 (binary), A001191 (concatenation of all squares), A134724 (cubes), A033308 (primes: Copeland-Erdős constant).

Sequence in context: A244978 A114864 A261025 * A185260 A275615 A249677

Adjacent sequences: A340204 A340205 A340206 * A340208 A340209 A340210

Keyword

nonn,base,cons

Author

M. F. Hasler, Jan 01 2021

Status

approved