A075772 Difference between the n-th perfect power and the closest perfect power.

3, 3, 1, 1, 7, 2, 2, 4, 4, 13, 15, 17, 19, 4, 3, 3, 16, 25, 20, 9, 9, 13, 13, 33, 19, 18, 18, 39, 41, 28, 17, 17, 47, 49, 51, 53, 55, 57, 59, 39, 24, 24, 65, 67, 69, 35, 35, 38, 75, 77, 79, 47, 36, 36, 85, 87, 23, 23, 68, 10, 10, 12, 95, 97, 99, 101, 40, 40, 65, 107, 100, 11, 11

Offset

1,1

Comments

Let {b(n)} be the sequence of perfect powers (A001597); then a(n) = min { b(n)-b(n-1), b(n+1)-b(n) }.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Formula

a(n) = min A053289({n, n-1}\{0}), where A053289(n) = A001597(n+1) - A001597(n). - M. F. Hasler, May 08 2018

Example

The perfect powers are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, etc. The 7th is 27. This is 2 larger than the 6th (25) and 5 smaller than the 8th (32). So a(7)=2.

Mathematica

pp = {-2, 1}; Do[ If[ !PrimeQ[n] && Apply[GCD, Last[ Transpose[ FactorInteger[n]]]] > 1, pp = Append[pp, n]], {n, 2, 10^4}]; Table[ Min[pp[[n + 1]] - pp[[n]], pp[[n + 2]] - pp[[n + 1]]], {n, 1, 75}]
perfPQ[n_]:=GCD@@FactorInteger[n][[All, 2]]>1; Join[{3, 3}, Min[ Differences[ #]]&/@Partition[Select[Range[5000], perfPQ], 3, 1]] (* Harvey P. Dale, May 04 2021 *)

Prog

(PARI) for(n=L=3+P=-2, 99, ispower(n)&&print1(min(-P+P=L, -L+L=n)", ")) \\ Note: ispower(1)=0. - M. F. Hasler, May 08 2018
(Python)
from sympy import mobius, integer_nthroot
def A075772(n):
    if n == 1: return 3
    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())))
    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, n-1, n-1)
    b = bisection(lambda x:f(x)+1, a, a)
    return min(b-a, bisection(lambda x:f(x)+2, b, b)-b) # Chai Wah Wu, Sep 09 2024

Crossrefs

Cf. A001597, A053289, A075773.

Sequence in context: A380518 A270273 A026515 * A142157 A119608 A375849

Adjacent sequences: A075769 A075770 A075771 * A075773 A075774 A075775

Keyword

nonn

Author

Neil Fernandez, Oct 09 2002

Extensions

More terms from Robert G. Wilson v and John W. Layman, Oct 10 2002

Status

approved