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A051634
Strong primes: prime(k) > (prime(k-1) + prime(k+1))/2.
42
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
1,1
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
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.
FORMULA
Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 1/2. - Alain Rocchelli, Mar 17 2024
EXAMPLE
11 belongs to the sequence because 11 > (7 + 13)/2.
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]]]]
PROG
(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
(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
(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
CROSSREFS
Subsequence of A178943.
Cf. A225493 (multiplicative closure), A131499 (subsequence).
Sequence in context: A240095 A105886 A225493 * A038918 A220293 A166307
KEYWORD
nice,nonn
AUTHOR
Felice Russo, Nov 15 1999
STATUS
approved