A076432 Perfect powers for which the three closest perfect powers are smaller.

36, 144, 2209, 6436369, 312079766881, 328081510656, 11305787558464, 62854912315881, 79723540870416, 4550858480922601, 11435943732416784, 3109406220195722500, 6258210474706101136, 7596357574791306304, 21016258678615763761, 32645304184825666489

Offset: 1

Author

Neil Fernandez, Oct 10 2002

Keywords

• nonn

Examples

			The three closest perfect powers to 36 are 32 (difference = 4), 27 (difference = 9) and 25 (difference = 11). The fourth closest is 49 (difference = 13). 32, 27 and 25 are smaller than 36, so 36 is in the sequence.
		

Crossrefs

Cf. A001597, A053289, A075772, A076431, A076433.

Programs

Python

from itertools import count, islice
from sympy import mobius, integer_nthroot
def A076432_gen(): # generator of terms
    def f(x): return int(x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    a = bisection(f)
    b = bisection(lambda x:f(x)+1,a,a)
    c = bisection(lambda x:f(x)+2,b,b)
    d = bisection(lambda x:f(x)+3,c,c)
    for i in count(4):
        e = bisection(lambda x:f(x)+i,d,d)
        if d-a < e-d:
            yield d
        a,b,c,d=b,c,d,e
A076432_list = list(islice(A076432_gen(),5)) # Chai Wah Wu, Sep 09 2024

Extensions

More terms from Jud McCranie and Robert G. Wilson v, Oct 11 2002

a(5)-a(10) from Donovan Johnson, Sep 03 2008

a(11)-a(16) from Donovan Johnson, Aug 01 2013