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A002760 Squares and cubes. 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

REFERENCES

Bilu, Y. F., Catalan's Conjecture (After Mihilescu). Asterisque, No. 294, 1-26, 2004.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 68.

C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 236.

LINKS

Zak Seidov and Michael De Vlieger, Table of n, a(n) for n = 1..10443 (First 1000 terms from Zak Seidov)

Y. F. Bilu, Catalan Without Logarithmic Forms (after Bugeaud, Hanrot and Mihailescu), J. Théor. Nombres Bordeaux 17, 69-85, 2005.

T. Metsänkylä, Catalan's conjecture: another old Diophantine problem solved, Bull. Amer. Math. Soc. (NS), Vol. 41, No. 1 (2004), pp. 43-57.

P. Mihailescu, A Class Number Free Criterion for Catalan's Conjecture, J. Number Th. 99 225-231, 2003.

P. Mihailescu, Primary Cyclotomic Units and a Proof of Catalan's Conjecture, J. Reine angew. Math. 572 (2004): 167-195. MR 2076124.

P. Ribenboim, Catalan's Conjecture, Séminaire de Philosophie et Mathématiques, 6 (1994), p. 1-11.

P. Ribenboim, Catalan's Conjecture, Amer. Math. Monthly, Vol. 103(7) Aug-Sept 1996, pp. 529-538.

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A131799; union of A000290 and A000578.

First differences in A075052. [From Zak Seidov, May 10 2010]

Sequence in context: A246547 A195942 A125643 * A115651 A062559 A010417

Adjacent sequences:  A002757 A002758 A002759 * A002761 A002762 A002763

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified November 21 16:27 EST 2017. Contains 295003 sequences.