A340206 - OEIS

%I #12 Jan 02 2021 08:36:26

%S 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,

%T 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,

%U 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

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

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

%C 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.

%F c = 0.11610010241000010048910000001000456910000000010000141291000000000010...

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

%F 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.

%e The smallest square with 1, 2, 3, 4, ... digits is, respectively, 1, 16 = 4^2, 100 = 10^2, 1024 = 32^2, ....

%e Here we list the digits of these numbers: 1; 1, 6; 1, 0, 0; 1, 0, 2, 4; ...

%e As for the Champernowne and Copeland-Erdős constants, we can consider this as the decimal expansion of the real constant 0.116100102410000100489...

%o (PARI) concat([digits(ceil(10^((k-1)/2))^2)|k<-[1..14]]) \\ as seq. of digits

%o c(N=20)=sum(k=1,N,.1^(k*(k+1)/2)*ceil(10^((k-1)/2))^2) \\ as constant

%Y 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).

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

%K nonn,base,cons

%O 0,3

%A _M. F. Hasler_, Dec 31 2020