A002760 Squares and cubes.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

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

• Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 68.
• Clifford A. Pickover, The Math Book, Sterling, NY, 2009; see p. 236.

Links

• 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.

Crossrefs

Cf. A131799; union of A000290 and A000578.

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

Cf. A002117, A013661, A013664.

Programs

Magma

[n: n in [0..1600] | IsIntegral(n^(1/3)) or IsIntegral(n^(1/2))]; // Bruno Berselli, Feb 09 2016

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 *)

PARI

isok(n) = issquare(n) || ispower(n, 3); \\ Michel Marcus, Mar 29 2016

Python

from math import isqrt
from sympy import integer_nthroot
def A002760(n):
    def f(x): return n-1+x+integer_nthroot(x,6)[0]-integer_nthroot(x,3)[0]-isqrt(x)
    m, k = n-1, f(n-1)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 09 2024

Formula

Sum_{n>=2} 1/a(n) = zeta(2) + zeta(3) - zeta(6). - Amiram Eldar, Dec 19 2020