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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]

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

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<p+r, print1(q", ")); p=q; q=r) \\ Charles R Greathouse IV, Jul 25 2011

(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

CROSSREFS

Cf. A006562, A051634, A229832.

Subsequence of A178943; A225495 (multiplicative closure).

Sequence in context: A105399 A133261 A113911 * A136009 A216098 A310262

Adjacent sequences:  A051632 A051633 A051634 * A051636 A051637 A051638

KEYWORD

nice,nonn,easy

AUTHOR

Felice Russo, Nov 15 1999

EXTENSIONS

More terms from James A. Sellers

STATUS

approved

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Last modified December 14 17:31 EST 2019. Contains 329979 sequences. (Running on oeis4.)