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

3, 19, 349, 2909, 15377, 128983, 1319411, 17797519, 94097539, 6927837559, 48486712787, 968068681519, 1472840004019, 129001208165719

Offset: 1

Author

Jonathan Sondow, Oct 13 2013

Keywords

• nonn
• more

Comments

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

Examples

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

Links

• 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

Crossrefs

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

Formula

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

Extensions

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

a(14) from Giovanni Resta, Apr 19 2016