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A125643 Squares and cubes (with repetition).

%I A125643 #27 Dec 20 2024 10:18:28
%S A125643 0,0,1,1,4,8,9,16,25,27,36,49,64,64,81,100,121,125,144,169,196,216,
%T A125643 225,256,289,324,343,361,400,441,484,512,529,576,625,676,729,729,784,
%U A125643 841,900,961,1000,1024,1089,1156,1225,1296,1331,1369,1444,1521,1600,1681
%N A125643 Squares and cubes (with repetition).
%C A125643 Repeating terms are sixth powers: 0,1,64,729,... (A001014).
%C A125643 For numbers not appearing as a difference between a square and an adjacent cube in this list, see A054504 and A081121.
%H A125643 Zak Seidov, <a href="/A125643/b125643.txt">Table of n, a(n) for n = 1..1000.</a>
%t A125643 m=1681;cm=Floor[m^(1/3)];sm=Floor[Sqrt[m]];s=Range[0,sm]^2;c=Range[0,cm]^3;Sort[Join[s,c]] (* _James C. McMahon_, Dec 20 2024 *)
%o A125643 (Python)
%o A125643 from math import isqrt
%o A125643 from sympy import integer_nthroot
%o A125643 def A125643(n):
%o A125643     if n <= 4: return n-1>>1
%o A125643     def bisection(f,kmin=0,kmax=1):
%o A125643         while f(kmax) > kmax: kmax <<= 1
%o A125643         while kmax-kmin > 1:
%o A125643             kmid = kmax+kmin>>1
%o A125643             if f(kmid) <= kmid:
%o A125643                 kmax = kmid
%o A125643             else:
%o A125643                 kmin = kmid
%o A125643         return kmax
%o A125643     def f(x): return n-2+x-integer_nthroot(x,3)[0]-isqrt(x)
%o A125643     return bisection(f,n-2,n-2) # _Chai Wah Wu_, Oct 14 2024
%Y A125643 Cf. A002760 (squares and cubes (without repetitions)).
%Y A125643 Cf. A001014, A054504, A081121, A087285, A087286, A088017.
%K A125643 nonn
%O A125643 1,5
%A A125643 _Zak Seidov_, Oct 19 2006
%E A125643 Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jul 14 2007