

A038458


Decimal expansion of the solution to 127^x  113^x = 1. This is the smallest x such that q^x  p^x = 1 for two successive primes p, q.


5



5, 6, 7, 1, 4, 8, 1, 3, 0, 2, 0, 2, 0, 1, 7, 7, 1, 4, 6, 4, 6, 8, 4, 6, 8, 7, 5, 5, 3, 3, 4, 8, 2, 5, 6, 4, 5, 8, 6, 7, 9, 0, 2, 4, 9, 3, 8, 8, 6, 3, 8, 2, 0, 6, 8, 4, 0, 2, 8, 5, 2, 2, 1, 8, 2, 6, 8, 0, 6, 7, 6, 6, 3, 3, 8, 2, 7, 6, 9, 2, 1, 5, 0, 8, 8, 6, 9, 7, 3, 8, 5, 3, 6, 4, 2, 6, 4, 4
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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. 6991.
David LowryDuda, 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. 173176.
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^x113^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^x113^x1); d=0; for (n=0, 20000, x=(xd)*10; d=floor(x); write("b038458.txt", n, " ", d)); \\ Harry J. Smith, Apr 13 2009


CROSSREFS

Sequence in context: A214681 A019978 A030178 * A284361 A267017 A021642
Adjacent sequences: A038455 A038456 A038457 * A038459 A038460 A038461


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



