A340219 Constant whose decimal expansion is the concatenation of the smallest n-digit prime A003617(n), for n = 1, 2, 3, ...

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, 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, 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

Offset: 0

Author

M. F. Hasler, Jan 01 2021

Keywords

• nonn
• base
• cons

Comments

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

Examples

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

Crossrefs

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).

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

Programs

Mathematica

Flatten[Table[IntegerDigits[NextPrime[10^n]],{n,0,20}]] (* Harvey P. Dale, Mar 29 2024 *)

PARI

concat([digits(nextprime(10^k))|k<-[0..14]]) \\ as seq. of digits
c(N=20)=sum(k=1,N,.1^(k*(k+1)/2)*nextprime(10^(k-1))) \\ as constant

Formula

c = 0.21110110091000710000310000031000001910000000710000000071000000001...

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

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