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A075052 Write down squares and cubes in order, as in A002760. The sequence gives the first differences between terms.

%I A075052 #17 Aug 09 2024 10:07:20
%S A075052 1,3,4,1,7,9,2,9,13,15,17,19,21,4,19,25,27,20,9,31,33,35,19,18,39,41,
%T A075052 43,28,17,47,49,51,53,55,57,59,61,39,24,65,67,69,71,35,38,75,77,79,81,
%U A075052 47,36,85,87,89,91,81,12,95,97,99,101,103,40,65,107,109,111,113,115,11
%N A075052 Write down squares and cubes in order, as in A002760. The sequence gives the first differences between terms.
%C A075052 Ordered sequence of squares and cubes is as in A002760.
%H A075052 Alois P. Heinz, <a href="/A075052/b075052.txt">Table of n, a(n) for n = 1..20000</a>
%F A075052 a(n) = A002760(n+1) - A002760(n).
%e A075052 First sorted squares and cubes are: 0, 1, 4, 8, 9, 16, 25, 27, 36, 49, 64, 81, 100, hence a(1) = 1 - 0 = 1, a(10) = 64 - 49 = 15, a(12) = 100 - 81 = 19.
%t A075052 Module[{nn=70,sq,cb},sq=Range[0,nn]^2;cb=Select[Range[nn]^3,#<=sq[[-1]]&];Join[sq,cb]//Union//Differences] (* _Harvey P. Dale_, Jun 11 2024 *)
%o A075052 (PARI) lista(nn) = {vec = vector(nn, i, i-1); pp = select(i->(issquare(i) || (ispower(i, 3))), vec); for (i=1, #pp-1, print1(pp[i+1] - pp[i], ", "););} \\ _Michel Marcus_, Oct 03 2013
%o A075052 (Python)
%o A075052 from math import isqrt
%o A075052 from sympy import integer_nthroot
%o A075052 def A075052(n):
%o A075052     def f(x): return n-1+x+integer_nthroot(x,6)[0]-(b:=integer_nthroot(x,3)[0])-(a:=isqrt(x)), a, b
%o A075052     m = n-1
%o A075052     k, a, b = f(n-1)
%o A075052     while m != k:
%o A075052         m = k
%o A075052         k, a, b = f(k)
%o A075052     return min((a+1)**2,(b+1)**3)-m # _Chai Wah Wu_, Aug 09 2024
%Y A075052 Cf. A002760.
%K A075052 nonn
%O A075052 1,2
%A A075052 _Zak Seidov_, Oct 07 2002