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A061109 a(1) = 1; a(n) = smallest number such that the concatenation a(1)a(2)...a(n) is an n-th power.

%I A061109 #18 Aug 15 2025 23:18:15
%S A061109 1,6,6375,34623551127976881,
%T A061109 18860302374385155610185422853070042488899966126368559233360607121925651097253827765970857
%N A061109 a(1) = 1; a(n) = smallest number such that the concatenation a(1)a(2)...a(n) is an n-th power.
%C A061109 Is this sequence infinite? - _Charles R Greathouse IV_, Sep 19 2012
%C A061109 From _Robert Israel_, Oct 05 2020: (Start)
%C A061109 If 10^m > ((x+1)^(1/n)-(x+1/10)^(1/n))^(-n), where x is the concatenation a(1)...a(n-1), then a(n) < 10^m.
%C A061109 In particular, the sequence is infinite.
%C A061109 a(6) has 558 digits, a(7) has 4014 digits, and a(8) has 32783 digits. (End)
%D A061109 Amarnath Murthy, Exploring some new ideas on Smarandache type sets, functions and sequences, Smarandache Notions Journal Vol. 11, No. 1-2-3, Spring 2000.
%e A061109 a(1) = 1, a(1)a(2) = 16 = 4^2, a(1)a(2)a(3) = 166375 = 55^3, a(1)a(2)a(3)a(4) = 16637534623551127976881 = 359147^4.
%p A061109 ncat:= (a,b) -> a*10^(1+ilog10(b))+b:
%p A061109 f:= proc(n,x)
%p A061109   local z,d;
%p A061109   for d from 1  do
%p A061109     z:= ceil(((x+1/10)*10^d)^(1/n));
%p A061109     if z^n < (x+1)*10^d then return z^n - x*10^d fi
%p A061109   od
%p A061109 end proc:
%p A061109 R[1]:= 1: C:= 1:
%p A061109 for n from 2 to 6 do
%p A061109   R[n]:= f(n,C);
%p A061109   C:= ncat(C, R[n]);
%p A061109 od:
%p A061109 seq(R[i],i=1..6); # _Robert Israel_, Oct 05 2020
%K A061109 base,nonn
%O A061109 1,2
%A A061109 _Amarnath Murthy_, Apr 20 2001
%E A061109 Corrected and extended by Ulrich Schimke, Feb 08 2002
%E A061109 Offset corrected by _Robert Israel_, Oct 05 2020