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
Zhuorui He, Table of n, a(n) for n = 0..11324 (first 150 rows)
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...
As a triangle, in which row n contains the decimal expansion of the smallest n-digit cube:
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
...
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).
AUTHOR
M. F. Hasler, Dec 31 2020
STATUS
approved