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A076405
Next perfect power having the same least root of n-th perfect power, A001597.
2
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
1,2
COMMENTS
A025478(a(n)) = A025478(n); A001597(a(n)) = A025478(n)*A001597(n).
LINKS
Eric Weisstein's World of Mathematics, Perfect Powers.
EXAMPLE
. 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).
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 *)
PROG
(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
(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
CROSSREFS
Cf. A052410.
Sequence in context: A111231 A111307 A246549 * A099997 A053093 A175086
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Oct 09 2002
EXTENSIONS
More terms from Robert G. Wilson v, Aug 03 2008
STATUS
approved