A025478 Least roots of perfect powers (A001597).

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, 22, 2, 23, 24, 5, 26, 3, 28, 29, 30, 31, 10, 2, 33, 34, 35, 6, 11, 37, 38, 39, 40, 41, 12, 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

Offset

1,2

Links

Daniel Forgues, Table of n, a(n) for n=1..10000

Formula

a(n) = A052410(A001597(n)).

(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..n} 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 is no asymptotic expression for a(n) (just the average in (iii)). - Benoit Cloitre, Oct 14 2002

Example

a(5) = 2 because pp(5) = 16 = 2^4 (not 4^2 as we take the smallest base).

Mathematica

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}]]
(* Alternative: *)
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 *)

Prog

(Haskell)
a025478 n = a025478_list !! (n-1)  -- a025478_list defined in A001597.
-- Reinhard Zumkeller, Mar 11 2014
(Python)
from math import gcd
from sympy import mobius, integer_nthroot, factorint
def A025478(n):
    if n == 1: return 1
    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())))
    kmin, kmax = 1, 2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return integer_nthroot(kmax, gcd(*factorint(kmax).values()))[0] # Chai Wah Wu, Aug 13 2024
(PARI) lista(kmax) = {my(r, e); print1(1, ", "); for(k = 1, kmax, e = ispower(k, , &r); if(e > 0, print1(r, ", "))); } \\ Amiram Eldar, Sep 07 2024

Crossrefs

Cf. A052410 (least root), A001597 (perfect powers).

Cf. A025479 (largest exponents of perfect powers), A070228.

Sequence in context: A264662 A076403 A157987 * A084371 A025476 A347731

Adjacent sequences: A025475 A025476 A025477 * A025479 A025480 A025481

Keyword

easy,nonn

Author

David W. Wilson

Extensions

Definition edited and cross-reference added by Daniel Forgues, Mar 10 2009

Status

approved