%I #24 Aug 14 2024 01:56:19

%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

%o (Python)

%o from math import gcd

%o from sympy import mobius, integer_nthroot, factorint

%o def A025478(n):

%o if n == 1: return 1

%o def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))

%o kmin, kmax = 1,2

%o while f(kmax) >= kmax:

%o kmax <<= 1

%o while True:

%o kmid = kmax+kmin>>1

%o if f(kmid) < kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o if kmax-kmin <= 1:

%o break

%o return integer_nthroot(kmax, gcd(*factorint(kmax).values()))[0] # _Chai Wah Wu_, Aug 13 2024

%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