A051635 Weak primes: prime(n) < (prime(n-1) + prime(n+1))/2.

3, 7, 13, 19, 23, 31, 43, 47, 61, 73, 83, 89, 103, 109, 113, 131, 139, 151, 167, 181, 193, 199, 229, 233, 241, 271, 283, 293, 313, 317, 337, 349, 353, 359, 383, 389, 401, 409, 421, 433, 443, 449, 463, 467, 491, 503, 509, 523, 547, 571, 577, 601, 619, 643, 647

Offset: 1

Author

Felice Russo, Nov 15 1999

Keywords

• nice
• nonn
• easy

Comments

Primes prime(n) such that prime(n)-prime(n-1) < prime(n+1)-prime(n). - Juri-Stepan Gerasimov, Jan 01 2011

a(n) < A051634(n). a(n) ~ 2*prime(n). - Thomas Ordowski, Jul 25 2012

The inequality above is false. The least counterexample is a(19799) = 496291 > A051634(19799) = 496283. - Amiram Eldar, Nov 26 2023

Erdős called a weak prime an "early prime." He conjectured that there are infinitely many consecutive pairs of early primes, and offered $100 for a proof and $25000 for a disproof (Kuperberg 1992). See A229832 for a stronger conjecture. - Jonathan Sondow, Oct 13 2013

Examples

			7 belongs to the sequence because 7 < (5+11)/2.
		

References

• A. Murthy, Smarandache Notions Journal, Vol. 11 N. 1-2-3 Spring 2000

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• 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. A006562, A051634, A229832.

Subsequence of A178943.

Cf. A225495 (multiplicative closure).

Programs

Haskell

a051635 n = a051635_list !! (n-1)
a051635_list = g a000040_list where
   g (p:qs@(q:r:ps)) = if 2 * q < (p + r) then q : g qs else g qs
-- Reinhard Zumkeller, May 09 2013

Mathematica

Transpose[Select[Partition[Prime[Range[10^2]], 3, 1], #[[2]]<(#[[1]]+#[[3]])/2 &]][[2]] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
p=Prime[Range[200]]; p[[Flatten[1+Position[Sign[Differences[p, 2]], 1]]]]

PARI

p=2;q=3;forprime(r=5,1e3,if(2*qCharles R Greathouse IV, Jul 25 2011

Formula

a(1) = A229832(1). - Jonathan Sondow, Oct 13 2013

Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 1/2. - Alain Rocchelli, Mar 17 2024

Extensions

More terms from James Sellers