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A051634
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Strong primes: prime(k) > (prime(k-1) + prime(k+1))/2.
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42
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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
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OFFSET
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1,1
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COMMENTS
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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 primepuzzles.net - Conjecture 91] -Alain Rocchelli,Nov 14 2022
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REFERENCES
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A. Murthy, Smarandache Notions Journal, Vol. 11 N. 1-2-3 Spring 2000.
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LINKS
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FORMULA
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Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 1/2. -Alain Rocchelli,Mar 17 2024
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EXAMPLE
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11 belongs to the sequence because 11 > (7 + 13)/2.
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MAPLE
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q:= n-> isprime(n) and 2*n>prevprime(n)+nextprime(n):
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MATHEMATICA
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p=Prime[Range[200]]; p[[Flatten[1+Position[Sign[Differences[p, 2]], -1]]]]
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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nice,nonn
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AUTHOR
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STATUS
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approved
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