A340208 Constant whose decimal expansion is the concatenation of the smallest n-digit cube A061434(n), for n = 1, 2, 3, ...

1, 2, 7, 1, 2, 5, 1, 0, 0, 0, 1, 0, 6, 4, 8, 1, 0, 3, 8, 2, 3, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 7, 7, 6, 9, 6, 1, 0, 0, 5, 4, 4, 6, 2, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 7, 8, 7, 3, 8, 7, 5, 1, 0, 0, 0, 2, 6, 5, 7, 7, 2, 8, 8, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset

0,2

Comments

Every third smallest n-digit cube (i.e., for n = 3k + 1, k >= 0) is 10^k, which explains the chunks of (1,0,...,0), cf. formula.

The terms of sequence A215692 converge to this sequence of digits, and to this constant, up to powers of 10.

Links

Table of n, a(n) for n=0..91.

Formula

c = 0.12712510001064810382310000001007769610054462510000000001000787387510002657...

= Sum_{k >= 1} 10^(-k(k+1)/2)*ceiling(10^((k-1)/3))^2

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

a(k) = 0 for -3n(3n+1)/2 > k > -(3n+1)(3n+2)/2, n >= 0.

Example

The smallest cube with 1, 2, 3, 4, ... digits is, respectively, 1, 27 = 3^3, 125 = 5^3, 1000 = 10^3, .... Here we list the sequence of digits of these numbers: 1; 2, 7; 1, 2, 5; 1, 0, 0, 0; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.1271251000106481...

Prog

(PARI) concat([digits(ceil(10^((k-1)/3))^3)|k<-[1..14]]) \\ as seq. of digits
c(N=12)=sum(k=1, N, .1^(k*(k+1)/2)*ceil(10^((k-1)/2))^2) \\ as constant

Crossrefs

Cf. A061434 (smallest n-digit cube), A215692 (has this as "limit"), A340209 (same with largest n-digit cubes, limit of A340115), A340206 (same for squares, limit of A215689), A340219 (same for primes, limit of A215641), 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).

Sequence in context: A081705 A371748 A324329 * A201889 A145057 A344930

Adjacent sequences: A340205 A340206 A340207 * A340209 A340210 A340211

Keyword

nonn,base,cons

Author

M. F. Hasler, Dec 31 2020

Status

approved