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A002760
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Squares and cubes.
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10
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0, 1, 4, 8, 9, 16, 25, 27, 36, 49, 64, 81, 100, 121, 125, 144, 169, 196, 216, 225, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849
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OFFSET
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1,3
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COMMENTS
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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.
Subsequence of A022549. - Reinhard Zumkeller, Jul 17 2010
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REFERENCES
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Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 68.
Clifford A. Pickover, The Math Book, Sterling, NY, 2009; see p. 236.
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LINKS
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Zak Seidov and Michael De Vlieger, Table of n, a(n) for n = 1..10443 (First 1000 terms from Zak Seidov)
Yuri F. Bilu, Catalan's Conjecture (After Mihilescu), Astérisque, No. 294, 1-26, 2004.
Yuri F. Bilu, Catalan Without Logarithmic Forms (after Bugeaud, Hanrot and Mihailescu), J. Théor. Nombres Bordeaux 17, 69-85, 2005.
David Masser, Alan Baker, arXiv:2010.10256 [math.HO], 2020. See p. 4.
Tauno Metsänkylä, Catalan's conjecture: another old Diophantine problem solved, Bull. Amer. Math. Soc. (NS), Vol. 41, No. 1 (2004), pp. 43-57.
Preda Mihǎilescu, A Class Number Free Criterion for Catalan's Conjecture, J. Number Th. 99 225-231, 2003.
Preda Mihǎilescu, Primary Cyclotomic Units and a Proof of Catalan's Conjecture, J. Reine angew. Math. 572 (2004): 167-195. MR 2076124.
Paulo Ribenboim, Catalan's Conjecture, Séminaire de Philosophie et Mathématiques, 6 (1994), p. 1-11.
Paulo Ribenboim, Catalan's Conjecture, Amer. Math. Monthly, Vol. 103(7) Aug-Sept 1996, pp. 529-538.
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FORMULA
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Sum_{n>=2} 1/a(n) = zeta(2) + zeta(3) - zeta(6). - Amiram Eldar, Dec 19 2020
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MATHEMATICA
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nMax=2000; Union[Range[0, nMax^(1/2)]^2, Range[0, nMax^(1/3)]^3] (* Vladimir Joseph Stephan Orlovsky, Apr 11 2011 *)
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 *)
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PROG
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(MAGMA) [n: n in [0..1600] | IsIntegral(n^(1/3)) or IsIntegral(n^(1/2))]; // Bruno Berselli, Feb 09 2016
(PARI) isok(n) = issquare(n) || ispower(n, 3); \\ Michel Marcus, Mar 29 2016
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CROSSREFS
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Cf. A131799; union of A000290 and A000578.
First differences in A075052. [From Zak Seidov, May 10 2010]
Cf. A002117, A013661, A013664.
Sequence in context: A246547 A195942 A125643 * A115651 A062559 A010417
Adjacent sequences: A002757 A002758 A002759 * A002761 A002762 A002763
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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