A264733 - OEIS

A264733

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

%I #51 Jan 16 2021 02:51:19

%S 4,9,13,31556,4433200001,7330164793357114944,

%T 364233003001227343654904892703798707409,

%U 30558883460500823396683989630832748682356643682219859233661160618544138815441

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

%H Robert Israel, <a href="/A264733/b264733.txt">Table of n, a(n) for n = 1..10</a>

%H 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 a[1]:= 4: C:= 4:

%p for n from 2 to 9 do

%p looking:= true;

%p for d from 1 while looking do

%p L:= 10^d*C + 10^(d-1);

%p U:= 10^d*C + 10^d - 1;

%p p:= 1;

%p while p < ilog2(U) do

%p p:= nextprime(p);

%p Lp:= ceil(L^(1/p));

%p Up:= floor(U^(1/p));

%p while not (Lp::integer and Up::integer) do

%p Digits:= 2*Digits;

%p Lp:= eval(Lp);

%p Up:= eval(Up);

%p od;

%p if Lp <= Up then

%p Cp:= Lp^p;

%p a[n]:= Cp - 10^d*C;

%p C:= Cp;

%p looking:= false;

%p break

%p fi

%p od

%p od:

%p seq(a[i],i=1..9); # _Robert Israel_, Nov 27 2015

%t 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 (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 Cf. A001597(perfect powers), A051671, A061109, A061110, A261696, A264738, A264776, A264777, A264804, A264842, A264848, A264849.

%K nonn,base,hard

%O 1,1

%A _Anders Hellström_, Nov 22 2015

%E a(5)-a(8) from _Jon E. Schoenfield_, Nov 22 2015