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 A051635 Weak primes: prime(n) < (prime(n-1) + prime(n+1))/2. 48
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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] ErdÅ‘s 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 REFERENCES A. Murthy, Smarandache Notions Journal, Vol. 11 N. 1-2-3 Spring 2000 LINKS T. D. Noe, Table of n, a(n) for n = 1..1000 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 FORMULA a(1) = A229832(1). - Jonathan Sondow, Oct 13 2013 EXAMPLE 7 belongs to the sequence because 7 < (5+11)/2. 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]]]] PROG (PARI) p=2; q=3; forprime(r=5, 1e3, if(2*q

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Last modified October 20 16:11 EDT 2019. Contains 328268 sequences. (Running on oeis4.)