A025479 -id:A025479 - cp's OEIS frontend

A157986 Largest exponents of perfect powers (m^k where m is an integer and k >= 2) multiplied by -1 when base m is prime (m^k thus a prime power).

2, -2, -3, -2, -4, -2, -3, -5, 2, -2, -6, -4, 2, -2, -3, -7, 2, -2, 2, 3, 2, -5, -8, -2, 2, -3, -2, 2, 2, 2, -9, -2, 2, -4, 2, -6, 2, -2, 2, -2, 3, -10, 2, 2, 2, 4, -3, -2, 2, 2, 2, -2, 3, 2, -2, 2, 2, -11, 2, -7, -3, -2, 2, -4, 2, 2, 2, 3, -2, 2, 2, -5, 2, 2, 2, 3, -2, 2, -2, 2, 2, -12, 2, 2

Offset: 1

Daniel Forgues, Mar 10 2009

sign

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

Cf. A001597 (perfect powers), A025479 (largest exponents of perfect powers).
Cf. A025478 (least roots of perfect powers).
Cf. A157985.

a(n) = {k}_n * (-1)^(Pi(m) - Pi(m-1)) where {k}_n is the exponent of {m^k}_n (the n-th perfect power with positive integer base m corresponding to largest integer exponent k) and Pi(m) is the prime counting function evaluated at m.

a(n) = A025479(n) * (-1)^{Pi(m) - Pi(m-1)}, with m = A001597(n)^(1/(A025479(n))).

A076405 Next perfect power having the same least root of n-th perfect power, A001597.

Original entry on oeis.org

1, 8, 16, 27, 32, 125, 81, 64, 216, 343, 128, 243, 1000, 1331, 625, 256, 1728, 2197, 2744, 1296, 3375, 729, 512, 4913, 5832, 2401, 6859, 8000, 9261, 10648, 1024, 12167, 13824, 3125, 17576, 2187, 21952, 24389, 27000, 29791, 10000, 2048, 35937, 39304

Offset: 1

Reinhard Zumkeller, Oct 09 2002

nonn

A025478(a(n)) = A025478(n); A001597(a(n)) = A025478(n)*A001597(n).

			.   n  | A001597(n) | A025478(n)^A025479(n) |  a(n)
. -----+------------+-----------------------+---------------------------
.  13  |       100  |         10^2          | 1000 = 10^3 = A001597(41)
.  14  |       121  |         11^2          | 1331 = 11^3 = A001597(47)
.  15  |       125  |          5^3          |  625 =  5^4 = A001597(34)
.  16  |       128  |          2^7          |  256 =  2^8 = A001597(23)
.  17  |       144  |         12^2          | 1728 = 12^3 = A001597(54).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Perfect Powers.

Cf. A052410.

a076405 n = a076405_list !! (n-1)
a076405_list = 1 : f (tail $ zip a001597_list a025478_list) where
   f ((p, r) : us) = g us where
     g ((q, r') : vs) = if r' == r then q : f us else g vs
-- Reinhard Zumkeller, Mar 11 2014

ppQ[n_] := GCD @@ Last /@ FactorInteger@# > 1; f[n_] := Block[{fi = Transpose@ FactorInteger@ n}, fi2 = fi[[2]]; Times @@ (fi[[1]]^(fi[[2]] (1 + 1/GCD @@ fi[[2]])))]; lst = Join[{1}, Select[ Range@ 1848, ppQ@# &]]; f /@ lst (* Robert G. Wilson v, Aug 03 2008 *)

from math import gcd
from sympy import mobius, integer_nthroot, factorint
def A076405(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 kmax*integer_nthroot(kmax, gcd(*factorint(kmax).values()))[0] # Chai Wah Wu, Aug 13 2024

More terms from Robert G. Wilson v, Aug 03 2008

A340643 Numbers k such that the two perfect powers immediately adjacent to k^2 both have exponents greater than 2.

Original entry on oeis.org

2, 3, 5, 15, 26, 46, 82, 89, 90, 129, 323, 362, 401, 420, 610, 624, 840, 2024, 2703, 2808, 6888, 12099, 15963, 19320, 24650, 29930, 33490, 36482, 39203, 45795, 47523, 52440, 66050, 69168, 83408, 94248, 94863, 103683, 114284, 164399, 185364, 206442, 222785, 227530, 229180

Offset: 1

Hugo Pfoertner, Jan 14 2021

nonn

Within the range of the data, a(n)^2 = A340642(n), i.e., no 3 immediately consecutive perfect powers x^p1, y^p2, z^p3 with min (p1, p2, p3) > 2 are seen. Is there a counterexample?

David A. Corneth, Table of n, a(n) for n = 1..611 (first 181 terms from Hugo Pfoertner)

Cf. A000290, A001597, A025479, A076467, A097054, A111245, A153158, A340642, A340700, A340701.

a340643(limit)={my(p2=999, p1=2, n2=1, n1=4); for(n=5, limit, my(p0=ispower(n)); if(p0>1, if(issquare(n1)&p2>2&p0>2, print1(sqrtint(n1),", ")); n2=n1; n1=n; p2=p1; p1=p0))};
a340643(10^8)

upto(n) = {n *= n; my(v = List(), res = List([2])); for(i = 2, sqrtnint(n, 3), for(e = 3, logint(n, i), listput(v, i^e) ); ); listsort(v, 1); for(i = 1, #v - 1, if(sqrtint(v[i]) + 1 == sqrtint(v[i+1]) - issquare(v[i+1]), listput(res, sqrtint(v[i+1]-issquare(v[i+1]))); ) ); res }

More terms from David A. Corneth, Jan 14 2021

A157987 Smallest roots m of perfect powers (m^k where m is an integer and k >= 2) multiplied by -1 when m is prime (m^k thus a prime power).

Original entry on oeis.org

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

Offset: 1

Daniel Forgues, Mar 10 2009, Mar 14 2009

sign

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

Cf. A157985 Perfect powers (m^k where m is an integer and k >= 2) multiplied by -1 when m is prime for largest k (m^k thus a prime power).
Cf. A157986 Largest exponents of perfect powers (m^k where m is an integer and k >= 2) multiplied by -1 when base m is prime (m^k thus a prime power).
Cf. A001597 Perfect powers: m^k where m is an integer and k >= 2.
Cf. A025479 Largest exponents of perfect powers (A001597).
Cf. A025478 Least roots of perfect powers (A001597).

a(n) = {m}_n * (-1)^{Pi(m) - Pi(m-1)}
where {m}_n is the smallest root of {m^k}_n (the n-th perfect power with positive integer base m corresponding to largest integer exponent k) and Pi(m) is the prime counting function evaluated at m.
a(n) = m * (-1)^{Pi(m) - Pi(m-1)}, with m = A025478(n) = {A001597(n)}^{1/{A025479(n)}}.

cp's OEIS Frontend

A157986 Largest exponents of perfect powers (m^k where m is an integer and k >= 2) multiplied by -1 when base m is prime (m^k thus a prime power).

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A076405 Next perfect power having the same least root of n-th perfect power, A001597.

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A340643 Numbers k such that the two perfect powers immediately adjacent to k^2 both have exponents greater than 2.

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A157987 Smallest roots m of perfect powers (m^k where m is an integer and k >= 2) multiplied by -1 when m is prime (m^k thus a prime power).

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