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A038458 Consider the equation q^x-p^x=1 where p,q are successive primes; solve for x; the smallest such x is 0.567148... which occurs when p=113, q=127. Sequence gives decimal expansion of this value of x. Sometimes called the Smarandache constant. 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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Generalizes Andrica's conjecture p(n+1)^(1/2)-p(n)^(1/2)<1 to p(n+1)^c-p(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. - John W. Nicholson, Dec 11 2013

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..20000

M. L. Perez, Five Smarandache Conjectures On Primes, Arizona State University, Special Collections.

F. Smarandache, Conjectures which Generalize Andrica's Conjecture, Octogon, Vol. 7, No. 1, 173-176, 1999.

Eric Weisstein's World of Mathematics, Andrica's Conjecture

Eric Weisstein's World of Mathematics, Smarandache Constant [broken link]

Eric Weisstein's World of Mathematics, Smarandache Constants

FORMULA

a(n) = solve(p(n+1)^x-p(n)^x < 1,x) with primes p(n+1) and p(n)

EXAMPLE

0.567148130202017714646846875533482564586790249388638206840285221826806766338276...

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)); } [From Harry J. Smith, Apr 13 2009]

CROSSREFS

Sequence in context: A214681 A019978 A030178 * A021642 A171423 A101288

Adjacent sequences:  A038455 A038456 A038457 * A038459 A038460 A038461

KEYWORD

nonn,cons

AUTHOR

M. I. Petrescu (mipetrescu(AT)yahoo.com)

STATUS

approved

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Last modified October 23 03:24 EDT 2014. Contains 248411 sequences.