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
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.
FORMULA
Sum_{n>=2} 1/a(n) = zeta(2) + zeta(3) - zeta(6). - Amiram Eldar, Dec 19 2020
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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved