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

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

Author

Reinhard Zumkeller, Oct 09 2002

Keywords

• nonn

Comments

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

Examples

			.   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).
		

Links

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

Crossrefs

Cf. A052410.

Programs

Haskell

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

Mathematica

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 *)

Python

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

Extensions

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