A340206 Constant whose decimal expansion is the concatenation of the smallest n-digit square A061432(n), for n = 1, 2, 3, ...

1, 1, 6, 1, 0, 0, 1, 0, 2, 4, 1, 0, 0, 0, 0, 1, 0, 0, 4, 8, 9, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 5, 6, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 4, 1, 2, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 4, 7, 9, 8, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 4, 9, 2, 8, 4, 1, 0, 0, 0, 0

Offset: 0

Author

M. F. Hasler, Dec 31 2020

Keywords

• nonn
• base
• cons

Comments

The terms of sequence A215689 have this as limit, digit-wise and as a constant, up to powers of 10.

Every other "smallest n-digit square" (i.e., for odd n = 2k + 1) is 10^k, which explains the chunks of (1,0,...,0), cf. formula.

Examples

			The smallest square with 1, 2, 3, 4, ... digits is, respectively, 1, 16 = 4^2, 100 = 10^2, 1024 = 32^2, ....
Here we list the digits of these numbers: 1; 1, 6; 1, 0, 0; 1, 0, 2, 4; ...
As for the Champernowne and Copeland-Erdős constants, we can consider this as the decimal expansion of the real constant 0.116100102410000100489...
		

Crossrefs

Cf. A061432 (smallest n-digit squares), A215689 (has this as "limit"), A340207 (same for largest n-digit squares), A340208 (same for cubes), A340219 (same for primes), A340221 (same for semiprimes).

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

Programs

PARI

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

Formula

c = 0.11610010241000010048910000001000456910000000010000141291000000000010...

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

a(-n(n+1)/2) = 1 for all n >= 0; a(k) = 0 for -n(n-1)/2 > k > -n(n+1)/2 with odd n.