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

Author

Neil Fernandez, Oct 09 2002

Keywords

• nonn

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) }.

Examples

			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.
		

Links

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

Crossrefs

Cf. A001597, A053289, A075773.

Programs

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 *)

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

Formula

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

Extensions

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