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A264733 a(n) is the smallest number > 1 such that the concatenation a(1)a(2)...a(n) is a perfect power.

%I A264733 #51 Jan 16 2021 02:51:19
%S A264733 4,9,13,31556,4433200001,7330164793357114944,
%T A264733 364233003001227343654904892703798707409,
%U A264733 30558883460500823396683989630832748682356643682219859233661160618544138815441
%N A264733 a(n) is the smallest number > 1 such that the concatenation a(1)a(2)...a(n) is a perfect power.
%H A264733 Robert Israel, <a href="/A264733/b264733.txt">Table of n, a(n) for n = 1..10</a>
%H A264733 Amarnath Murthy, <a href="https://web.archive.org/web/20150912233950/http://www.gallup.unm.edu/~smarandache/SN/ScArt5/ExploringNewIdeas.pdf">Exploring some new ideas on Smarandache type sets, functions and sequences</a>, Smarandache Notions Journal Vol. 11 N. 1-2-3 Spring 2000. p. 172 (breakup sequences).
%p A264733 a[1]:= 4: C:= 4:
%p A264733 for n from 2 to 9 do
%p A264733   looking:= true;
%p A264733   for d from 1 while looking do
%p A264733      L:= 10^d*C + 10^(d-1);
%p A264733      U:= 10^d*C + 10^d - 1;
%p A264733      p:= 1;
%p A264733      while p < ilog2(U) do
%p A264733       p:= nextprime(p);
%p A264733         Lp:= ceil(L^(1/p));
%p A264733         Up:= floor(U^(1/p));
%p A264733         while not (Lp::integer and Up::integer) do
%p A264733            Digits:= 2*Digits;
%p A264733            Lp:= eval(Lp);
%p A264733            Up:= eval(Up);
%p A264733         od;
%p A264733         if Lp <= Up then
%p A264733           Cp:= Lp^p;
%p A264733           a[n]:= Cp - 10^d*C;
%p A264733           C:= Cp;
%p A264733           looking:= false;
%p A264733           break
%p A264733         fi
%p A264733      od
%p A264733   od
%p A264733 od:
%p A264733 seq(a[i],i=1..9); # _Robert Israel_, Nov 27 2015
%t A264733 a = {}; Do[k = 2; While[! Or[# == 1, GCD @@ FactorInteger[#][[All, -1]] > 1] &@ FromDigits@ Flatten@ Join[#, IntegerDigits@ k], k++] &@ Map[IntegerDigits, a]; AppendTo[a, k], {i, 4}]; a (* _Michael De Vlieger_, Jan 23 2017 *)
%o A264733 (PARI) first(m)=my(s="4"); print1(4, ", "); for(i=2,m,n=1; while(!ispower(eval(concat(s,Str(n)))),n++); print1(n, ", "); s=concat(s,Str(n)))
%Y A264733 Cf. A001597(perfect powers), A051671, A061109, A061110, A261696, A264738, A264776, A264777, A264804, A264842, A264848, A264849.
%K A264733 nonn,base,hard
%O A264733 1,1
%A A264733 _Anders Hellström_, Nov 22 2015
%E A264733 a(5)-a(8) from _Jon E. Schoenfield_, Nov 22 2015