A051634 Strong primes: prime(k) > (prime(k-1) + prime(k+1))/2.

11, 17, 29, 37, 41, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 163, 179, 191, 197, 223, 227, 239, 251, 269, 277, 281, 307, 311, 331, 347, 367, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499, 521, 541, 557, 569, 587, 599, 613, 617, 631, 641, 659, 673, 701

Offset: 1

Author

Felice Russo, Nov 15 1999

Keywords

• nice
• nonn

Comments

Prime(k) such that prime(k) - prime(k-1) > prime(k+1) - prime(k). - Juri-Stepan Gerasimov, Jan 01 2011

a(n) > A051635(n). - Thomas Ordowski, Jul 25 2012

The inequality above is false. The least counterexample is a(19799) = 496283 < A051635(19799) = 496291. - Amiram Eldar, Nov 26 2023

Conjecture: Limit_{N->oo} Sum_{n=1..N} (NextPrime(a(n))-a(n)) / a(N) = 1/4. [A heuristic proof is available at www.primepuzzles.net - Conjecture 91] - Alain Rocchelli, Nov 14 2022

A131499 is a subsequence. - Davide Rotondo, Oct 16 2023

Examples

			11 belongs to the sequence because 11 > (7 + 13)/2.
		

References

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• Carlos Rivera, Conjecture 91. A conjecture about strong primes, The Prime Puzzles & Problems Connection.

Crossrefs

Cf. A006562, A051635, A229832.

Subsequence of A178943.

Cf. A225493 (multiplicative closure), A131499 (subsequence).

Programs

Haskell

a051634 n = a051634_list !! (n-1)
a051634_list = f a000040_list where
   f (p:qs@(q:r:ps)) = if 2 * q > (p + r) then q : f qs else f qs
-- Reinhard Zumkeller, May 09 2013

Maple

q:= n-> isprime(n) and 2*n>prevprime(n)+nextprime(n):
select(q, [$3..1000])[];  # Alois P. Heinz, Jun 21 2023

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=5,1e4,if(2*q>p+r,print1(q", "));p=q;q=r) \\ Charles R Greathouse IV, Jul 19 2011

Python

from sympy import nextprime
def aupto(limit):
    alst, p, q, r = [], 2, 3, 5
    while q <= limit:
        if 2*q > p + r: alst.append(q)
        p, q, r = q, r, nextprime(r)
    return alst
print(aupto(701)) # Michael S. Branicky, Nov 17 2021

Formula

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