1,1

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.

I make the stronger conjecture that the sequence a(n) is infinite.

a(1) = A051635(1), a(2) = A054820(1), a(3) = A054824(1), a(4) = A054829(1), a(5) = A054835(1).

a(n) is the prime following A158939(n+1). [Follows from the definitions] - Chris Boyd, Mar 28 2015

Greg Kuperberg, The Erdos kitty: At least $9050 in prizes!, Newsgroups: rec.puzzles, sci.math, 1992. [Broken link]

Greg Kuperberg, The Erdos kitty: At least $9050 in prizes!, Newsgroups: rec.puzzles, sci.math, 1992. [Cached copy]

Wikipedia, Weak prime

a(n) = min{p(i): 2*p(i+j) < p(i+j-1) + p(i+j+1), j = 0,1,..,n-1}.

The primes 19 < (17+23)/2 and 23 < (19+29)/2 are the smallest pair of consecutive weak/early primes, so a(2) = 19.

Cf. A051634, A051635, A054820, A054824, A054829, A054835, A158939.

Sequence in context: A346840 A132876 A357956 * A219270 A071381 A195639

Adjacent sequences: A229829 A229830 A229831 * A229833 A229834 A229835

nonn,more

Jonathan Sondow, Oct 13 2013

a(6) corrected by and a(7)-a(13) from Giovanni Resta, Jan 16 2014

a(14) from Giovanni Resta, Apr 19 2016

approved