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

%I A340208 #19 Sep 04 2025 21:54:07
%S A340208 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,
%T A340208 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,
%U A340208 0,0,2,6,5,7,7,2,8,8,1,0,0,0,0,0,0,0,0,0,0,0,0,1
%N A340208 Constant whose decimal expansion is the concatenation of the smallest n-digit cube A061434(n), for n = 1, 2, 3, ...
%C A340208 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.
%C A340208 The terms of sequence A215692 converge to this sequence of digits, and to this constant, up to powers of 10.
%H A340208 Zhuorui He, <a href="/A340208/b340208.txt">Table of n, a(n) for n = 0..11324</a> (first 150 rows)
%F A340208 c = 0.12712510001064810382310000001007769610054462510000000001000787387510002657...
%F A340208   = Sum_{k >= 1} 10^(-k(k+1)/2)*ceiling(10^((k-1)/3))^2
%F A340208 a(-n(n+1)/2) = 1 for all n >= 2;
%F A340208 a(k) = 0 for -3n(3n+1)/2 > k > -(3n+1)(3n+2)/2, n >= 0.
%e A340208 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; ...
%e A340208 This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.1271251000106481...
%e A340208 As a triangle, in which row n contains the decimal expansion of the smallest n-digit cube:
%e A340208   1
%e A340208   2 7
%e A340208   1 2 5
%e A340208   1 0 0 0
%e A340208   1 0 6 4 8
%e A340208   1 0 3 8 2 3
%e A340208   1 0 0 0 0 0 0
%e A340208   1 0 0 7 7 6 9 6
%e A340208   ...
%o A340208 (PARI) concat([digits(ceil(10^((k-1)/3))^3)|k<-[1..14]]) \\ as seq. of digits
%o A340208 c(N=12)=sum(k=1,N,.1^(k*(k+1)/2)*ceil(10^((k-1)/2))^2) \\ as constant
%Y A340208 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).
%Y A340208 Cf. A033307 (Champernowne constant), A030190 (binary), A001191 (concatenation of all squares), A134724 (cubes), A033308 (primes: Copeland-Erdős constant).
%K A340208 nonn,base,cons,tabl,changed
%O A340208 0,2
%A A340208 _M. F. Hasler_, Dec 31 2020