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

%I A075772 #22 Sep 10 2024 00:23:56
%S A075772 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,
%T A075772 39,41,28,17,17,47,49,51,53,55,57,59,39,24,24,65,67,69,35,35,38,75,77,
%U A075772 79,47,36,36,85,87,23,23,68,10,10,12,95,97,99,101,40,40,65,107,100,11,11
%N A075772 Difference between the n-th perfect power and the closest perfect power.
%C A075772 Let {b(n)} be the sequence of perfect powers (A001597); then a(n) = min { b(n)-b(n-1), b(n+1)-b(n) }.
%H A075772 Harvey P. Dale, <a href="/A075772/b075772.txt">Table of n, a(n) for n = 1..1000</a>
%F A075772 a(n) = min A053289({n, n-1}\{0}), where A053289(n) = A001597(n+1) - A001597(n). - _M. F. Hasler_, May 08 2018
%e A075772 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.
%t A075772 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}]
%t A075772 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 *)
%o A075772 (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
%o A075772 (Python)
%o A075772 from sympy import mobius, integer_nthroot
%o A075772 def A075772(n):
%o A075772     if n == 1: return 3
%o A075772     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())))
%o A075772     def bisection(f,kmin=0,kmax=1):
%o A075772         while f(kmax) > kmax: kmax <<= 1
%o A075772         while kmax-kmin > 1:
%o A075772             kmid = kmax+kmin>>1
%o A075772             if f(kmid) <= kmid:
%o A075772                 kmax = kmid
%o A075772             else:
%o A075772                 kmin = kmid
%o A075772         return kmax
%o A075772     a = bisection(f,n-1,n-1)
%o A075772     b = bisection(lambda x:f(x)+1,a,a)
%o A075772     return min(b-a,bisection(lambda x:f(x)+2,b,b)-b) # _Chai Wah Wu_, Sep 09 2024
%Y A075772 Cf. A001597, A053289, A075773.
%K A075772 nonn
%O A075772 1,1
%A A075772 _Neil Fernandez_, Oct 09 2002
%E A075772 More terms from _Robert G. Wilson v_ and _John W. Layman_, Oct 10 2002