cp's OEIS Frontend

A229832 First term of smallest sequence of n consecutive weak primes.

%I A229832 #28 May 07 2016 21:43:48
%S A229832 3,19,349,2909,15377,128983,1319411,17797519,94097539,6927837559,
%T A229832 48486712787,968068681519,1472840004019,129001208165719
%N A229832 First term of smallest sequence of n consecutive weak primes.
%C A229832 Erdős called a weak prime A051635 an "early prime," defined to be one which is less than the arithmetic mean of the prime before it and the prime after it. He conjectured that there are infinitely many consecutive pairs of early primes, and offered $100 for a proof and $25000 for a disproof. See Kuperberg 1992.
%C A229832 I make the stronger conjecture that the sequence a(n) is infinite.
%C A229832 a(1) = A051635(1), a(2) = A054820(1), a(3) = A054824(1), a(4) = A054829(1), a(5) = A054835(1).
%C A229832 a(n) is the prime following A158939(n+1). [Follows from the definitions] - _Chris Boyd_, Mar 28 2015
%H A229832 Greg Kuperberg, <a href="http://www.math.niu.edu/~rusin/known-math/93_back/prizes.erd">The Erdos kitty: At least $9050 in prizes!</a>, Newsgroups: rec.puzzles, sci.math, 1992. [Broken link]
%H A229832 Greg Kuperberg, <a href="/A051635/a051635.erd.txt">The Erdos kitty: At least $9050 in prizes!</a>, Newsgroups: rec.puzzles, sci.math, 1992. [Cached copy]
%H A229832 Wikipedia, <a href="https://en.wikipedia.org/wiki/Weak_prime">Weak prime</a>
%F A229832 a(n) = min{p(i): 2*p(i+j) < p(i+j-1) + p(i+j+1), j = 0,1,..,n-1}.
%e A229832 The primes 19 < (17+23)/2 and 23 < (19+29)/2 are the smallest pair of consecutive weak/early primes, so a(2) = 19.
%Y A229832 Cf. A051634, A051635, A054820, A054824, A054829, A054835, A158939.
%K A229832 nonn,more
%O A229832 1,1
%A A229832 _Jonathan Sondow_, Oct 13 2013
%E A229832 a(6) corrected by and a(7)-a(13) from _Giovanni Resta_, Jan 16 2014
%E A229832 a(14) from _Giovanni Resta_, Apr 19 2016