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%I #19 Apr 25 2017 18:59:56
%S 1,2,2,3,2,5,3,2,6,7,2,3,10,11,5,2,12,13,14,6,15,3,2,17,18,7,19,20,21,
%T 22,2,23,24,5,26,3,28,29,30,31,10,2,33,34,35,6,11,37,38,39,40,41,12,
%U 42,43,44,45,2,46,3,13,47,48,7,50,51,52,14,53,54,55,5,56,57,58,15,59,60,61,62
%N Least roots of perfect powers (A001597).
%H Daniel Forgues, <a href="/A025478/b025478.txt">Table of n, a(n) for n=1..10000</a>
%F a(n) = A052410(A001597(n)).
%F (i) a(n) < n for n>2. (ii) a(n)/n is bounded and lim sup a(n)/n must be around 0.7. (iii) sum(k=1, k, a(k)) seems to be asymptotic to c*n^2 with c around 0.29. (iv) a(n) = 2 if n is in A070228 (proof seems self-evident), hence there's no asymptotic expression for a(n) (just the average in (iii)). - _Benoit Cloitre_, Oct 14 2002
%e a(5)=2 because pp(5)=16=2^4 (not 4^2 as we take the smallest base).
%t pp = Select[ Range[5000], Apply[GCD, Last[ Transpose[ FactorInteger[ # ]]]] > 1 &]; f[n_] := Block[{b = 2}, While[ !IntegerQ[ Log[b, pp[[n]]]], b++ ]; b]; Join[{1}, Table[ f[n], {n, 2, 80}]]
%t (* Second program: *)
%t Prepend[DeleteCases[#, 0], 1] &@ Table[If[Set[e, GCD @@ #[[All, -1]]] > 1, Power[n, 1/e], 0] &@ FactorInteger@ n, {n, 4000}] (* _Michael De Vlieger_, Apr 25 2017 *)
%o (Haskell)
%o a025478 n = a025478_list !! (n-1) -- a025478_list defined in A001597.
%o -- _Reinhard Zumkeller_, Mar 11 2014
%Y Cf. A052410 (least root), A001597 (perfect powers).
%Y Cf. A025479 (largest exponents of perfect powers).
%K easy,nonn
%O 1,2
%A _David W. Wilson_
%E Added cross-reference. Definition edited by _Daniel Forgues_, Mar 10 2009
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