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%I #50 Oct 20 2023 11:54:10
%S 0,1,4,8,9,16,25,27,36,49,64,81,100,121,125,144,169,196,216,225,256,
%T 289,324,343,361,400,441,484,512,529,576,625,676,729,784,841,900,961,
%U 1000,1024,1089,1156,1225,1296,1331,1369,1444,1521,1600,1681,1728,1764,1849
%N Squares and cubes.
%C Catalan's Conjecture states that 8 and 9 are the only pair of consecutive numbers in this sequence. The conjecture was established in 2003 by Mihilescu.
%C Subsequence of A022549. - _Reinhard Zumkeller_, Jul 17 2010
%D Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 68.
%D Clifford A. Pickover, The Math Book, Sterling, NY, 2009; see p. 236.
%H Michael De Vlieger, <a href="/A002760/b002760.txt">Table of n, a(n) for n = 1..10443</a> (first 1000 terms from Zak Seidov)
%H Yuri F. Bilu, <a href="http://www.numdam.org/book-part/SB_2002-2003__45__1_0/">Catalan's Conjecture (After Mihilescu)</a>, Astérisque, No. 294, 1-26, 2004.
%H Yuri F. Bilu, <a href="http://dx.doi.org/10.5802/jtnb.478">Catalan Without Logarithmic Forms (after Bugeaud, Hanrot and Mihailescu)</a>, J. Théor. Nombres Bordeaux 17, 69-85, 2005.
%H David Masser, <a href="https://arxiv.org/abs/2010.10256">Alan Baker</a>, arXiv:2010.10256 [math.HO], 2020. See p. 4.
%H Tauno Metsänkylä, <a href="http://dx.doi.org/10.1090/S0273-0979-03-00993-5">Catalan's conjecture: another old Diophantine problem solved</a>, Bull. Amer. Math. Soc. (NS), Vol. 41, No. 1 (2004), pp. 43-57.
%H Preda Mihǎilescu, <a href="http://dx.doi.org/10.1016/S0022-314X(02)00101-4">A Class Number Free Criterion for Catalan's Conjecture</a>, J. Number Th. 99 225-231, 2003.
%H Preda Mihǎilescu, <a href="http://dx.doi.org/10.1515/crll.2004.048">Primary Cyclotomic Units and a Proof of Catalan's Conjecture</a>, J. Reine angew. Math. 572 (2004): 167-195. MR 2076124.
%H Paulo Ribenboim, <a href="http://www.numdam.org/item?id=SPHM_1994___6_A1_0">Catalan's Conjecture</a>, Séminaire de Philosophie et Mathématiques, 6 (1994), p. 1-11.
%H Paulo Ribenboim, <a href="http://www.jstor.org/stable/2974663">Catalan's Conjecture</a>, Amer. Math. Monthly, Vol. 103(7) Aug-Sept 1996, pp. 529-538.
%F Sum_{n>=2} 1/a(n) = zeta(2) + zeta(3) - zeta(6). - _Amiram Eldar_, Dec 19 2020
%t nMax=2000;Union[Range[0,nMax^(1/2)]^2,Range[0,nMax^(1/3)]^3] (* _Vladimir Joseph Stephan Orlovsky_, Apr 11 2011 *)
%t nxt[n_] := Min[ Floor[1 + Sqrt[n]]^2, Floor[1 + n^(1/3)]^3]; NestList[ nxt, 0, 55] (* _Robert G. Wilson v_, Aug 16 2014 *)
%o (Magma) [n: n in [0..1600] | IsIntegral(n^(1/3)) or IsIntegral(n^(1/2))]; // _Bruno Berselli_, Feb 09 2016
%o (PARI) isok(n) = issquare(n) || ispower(n, 3); \\ _Michel Marcus_, Mar 29 2016
%Y Cf. A131799; union of A000290 and A000578.
%Y First differences in A075052. [From _Zak Seidov_, May 10 2010]
%Y Cf. A002117, A013661, A013664.
%K nonn
%O 1,3
%A _N. J. A. Sloane_
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