A070228 -id:A070228 - cp's OEIS frontend

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

David W. Wilson

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

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

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

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

a025478 n = a025478_list !! (n-1)  -- a025478_list defined in A001597.
-- Reinhard Zumkeller, Mar 11 2014

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}]]
(* Second program: *)
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 *)

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

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

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

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

A188951 Number of perfect powers (A001597) < 2^n.

Original entry on oeis.org

0, 1, 1, 2, 4, 7, 10, 15, 22, 30, 41, 57, 81, 113, 155, 216, 298, 416, 582, 813, 1135, 1588, 2223, 3115, 4368, 6135, 8622, 12127, 17063, 24022, 33838, 47688, 67226, 94804, 133737, 188709, 266350, 376018, 530940, 749819, 1059096, 1496143, 2113801, 2986769

Offset: 0

T. D. Noe, Apr 20 2011

nonn

			For n=3, the perfect powers smaller than 2^3=8 are: 1 and 4. So a(3) = 2.

Amiram Eldar, Table of n, a(n) for n = 0..6643

Cf. A001597, A070228, A070428 (perfect powers not exceeding 10^n).

Join[{0,1}, Table[-Sum[MoebiusMu[x]*Floor[2^(n/x) - 1], {x, 2, n}], {n, 2, 50}]]

a(n) = sum(k=1, 2^n-1, (k==1) || ispower(k)); \\ Michel Marcus, Apr 11 2016

from sympy import mobius, integer_nthroot
def A188951(n): return int(sum(mobius(x)*(1-integer_nthroot(1<Chai Wah Wu, Aug 13 2024

a(n) = A070228(n) - 1 for n > 1. - Amiram Eldar, May 19 2022

A380337 Number of perfect powers (in A001597) that do not exceed primorial A002110(n).

Original entry on oeis.org

1, 1, 2, 7, 19, 63, 208, 802, 3344, 15576, 82368, 453834, 2743903, 17510668, 114616907, 785002449, 5711892439, 43861741799, 342522899289, 2803468693325, 23621594605383, 201819398349092, 1793794228847381, 16342173067958793, 154171432351500060, 1518411003599957803

Offset: 0

Michael De Vlieger, Jan 21 2025

nonn

In other words, A001597(a(n)) is the largest perfect power less than or equal to A002110(n).

			Let P = A002110 and let s = A001597.
a(0) = 1 since P(0) = 1, and the set s(1) = {1} contains k that do not exceed 1.
a(1) = 1 since P(1) = 2, and the set s(1) = {1} contains k <= 2.
a(2) = 2 since P(2) = 6, and the set s(1..2) = {1, 4} contains k <= 6.
a(3) = 7 since P(3) = 30, and the set s(1..7) = {1, 4, 8, 9, 16, 25, 27} contains k <= 30.
a(4) = 19 since P(4) = 210, and the set s(1..19) = {1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196} contains k <= 210, etc.

Michael De Vlieger, Table of n, a(n) for n = 0..632

Cf. A001597, A002110, A070228, A070428, A380254.

Map[1 - Sum[MoebiusMu[k]*Floor[#^(1/k) - 1], {k, 2, Floor[Log2[#]]}] &, FoldList[Times, 1, Prime[Range[30]]] ]

from sympy import primorial, mobius, integer_nthroot
def A380337(n):
    if n == 0: return 1
    p = primorial(n)
    return int(1-sum(mobius(k)*(integer_nthroot(p,k)[0]-1) for k in range(2,p.bit_length()))) # Chai Wah Wu, Jan 23 2025

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A025478 Least roots of perfect powers (A001597).

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A188951 Number of perfect powers (A001597) < 2^n.

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A380337 Number of perfect powers (in A001597) that do not exceed primorial A002110(n).

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