A340219 - OEIS

%I #8 Mar 29 2024 16:38:13

%S 2,1,1,1,0,1,1,0,0,9,1,0,0,0,7,1,0,0,0,0,3,1,0,0,0,0,0,3,1,0,0,0,0,0,

%T 1,9,1,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,1,9,1,0,

%U 0,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,0,3,9,1,0,0,0,0,0,0,0,0,0,0,0,3,7,1,0,0,0

%N Constant whose decimal expansion is the concatenation of the smallest n-digit prime A003617(n), for n = 1, 2, 3, ...

%C The terms of sequence A215641 converge to this sequence of digits, and to this constant, up to powers of 10.

%F c = 0.21110110091000710000310000031000001910000000710000000071000000001...

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

%F a(-n(n+1)/2) = 1 for all n >= 2, followed by increasingly more zeros.

%e The smallest prime with 1, 2, 3, 4, ... digits is, respectively, 2, 11, 101, 1009, .... Here we list the sequence of digits of these numbers: 2: 1, 1; 1, 0, 1; 1, 0, 0, 9; ...

%e This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.2111011009...

%t Flatten[Table[IntegerDigits[NextPrime[10^n]],{n,0,20}]] (* _Harvey P. Dale_, Mar 29 2024 *)

%o (PARI) concat([digits(nextprime(10^k))|k<-[0..14]]) \\ as seq. of digits

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

%Y Cf. A003617 (smallest n-digit prime), A215641 (has this as "limit"), A340206 (same for squares, limit of A215689), A340207 (similar, with largest n-digit squares, limit of A339978), A340208 (same for cubes, limit of A215692), A340209 (same with largest n-digit cube, limit of A340115), A340221 (same for 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