|
|
A051635
|
|
Weak primes: prime(n) < (prime(n-1) + prime(n+1))/2.
|
|
50
|
|
|
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
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
|
|
REFERENCES
|
A. Murthy, Smarandache Notions Journal, Vol. 11 N. 1-2-3 Spring 2000
|
|
LINKS
|
|
|
FORMULA
|
Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 1/2. - Alain Rocchelli, Mar 17 2024
|
|
EXAMPLE
|
7 belongs to the sequence because 7 < (5+11)/2.
|
|
MATHEMATICA
|
p=Prime[Range[200]]; p[[Flatten[1+Position[Sign[Differences[p, 2]], 1]]]]
|
|
PROG
|
(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
|
|
CROSSREFS
|
Cf. A225495 (multiplicative closure).
|
|
KEYWORD
|
nice,nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|