OFFSET
0,1
COMMENTS
Generalizes Andrica's conjecture prime(n+1)^(1/2) - prime(n)^(1/2) < 1 to prime(n+1)^c - prime(n)^c < 1 if c is less than this number.
Is this constant rational or irrational? I conjecture it is irrational. - Sukanto Bhattacharya (susant5au(AT)yahoo.com.au), Apr 28 2008
The first five digits are the same as the first five of A030178 = LambertW(1). - John W. Nicholson, Dec 11 2013
Although the description of the sequence defines it as "the smallest x" with a certain property, this is conjectured, not yet proven. Numerical evidence supports the conjecture. - Hal M. Switkay, Jun 02 2021
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..20000
Octavian Cira, Smarandache's conjecture on consecutive primes, International J. Math. Combin. 4 (2014), pp. 69-91.
David Lowry-Duda, A short note on gaps between powers of consecutive primes, arXiv:1709.07847 [math.NT], 2017.
M. L. Perez, Five Smarandache conjectures on primes, Arizona State University, Special Collections.
F. Smarandache, Conjectures which generalize Andrica's conjecture, arXiv:0707.2584 [math.GM], 2007; Octogon 7:1 (1999), pp. 173-176.
Eric Weisstein's World of Mathematics, Andrica's Conjecture
Eric Weisstein's World of Mathematics, Smarandache Constants
EXAMPLE
0.567148130202017714646846875533482564586790249388638206840285221826806766338276...
MATHEMATICA
RealDigits[x/.FindRoot[127^x-113^x==1, {x, 0.5}, WorkingPrecision->150]][[1]] (* Harvey P. Dale, Oct 24 2017 *)
PROG
(PARI) default(realprecision, 20080); x=solve(x=.5, .6, 127^x-113^x-1); d=0; for (n=0, 20000, x=(x-d)*10; d=floor(x); write("b038458.txt", n, " ", d)); \\ Harry J. Smith, Apr 13 2009
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
M. I. Petrescu (mipetrescu(AT)yahoo.com)
EXTENSIONS
Title improved, incorrect formula deleted, and other edits by M. F. Hasler, Jan 02 2015
STATUS
approved