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%I A051634 #70 Apr 09 2024 15:58:02
%S A051634 11,17,29,37,41,59,67,71,79,97,101,107,127,137,149,163,179,191,197,
%T A051634 223,227,239,251,269,277,281,307,311,331,347,367,379,397,419,431,439,
%U A051634 457,461,479,487,499,521,541,557,569,587,599,613,617,631,641,659,673,701
%N A051634 Strong primes: prime(k) > (prime(k-1) + prime(k+1))/2.
%C A051634 Prime(k) such that prime(k) - prime(k-1) > prime(k+1) - prime(k). - _Juri-Stepan Gerasimov_, Jan 01 2011
%C A051634 a(n) > A051635(n). - _Thomas Ordowski_, Jul 25 2012
%C A051634 The inequality above is false. The least counterexample is a(19799) = 496283 < A051635(19799) = 496291. - _Amiram Eldar_, Nov 26 2023
%C A051634 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
%C A051634 A131499 is a subsequence. - _Davide Rotondo_, Oct 16 2023
%D A051634 A. Murthy, Smarandache Notions Journal, Vol. 11 N. 1-2-3 Spring 2000.
%H A051634 Alois P. Heinz, <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL0EwNTE2MzQvYjA1MTYzNC50eHQ">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A051634 Carlos Rivera, <a href="/load/view.php?a=aHR0cHM6Ly93d3cucHJpbWVwdXp6bGVzLm5ldC9jb25qZWN0dXJlcy9jb25qXzA5MS5odG0">Conjecture 91. A conjecture about strong primes</a>, The Prime Puzzles & Problems Connection.
%F A051634 Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 1/2. - _Alain Rocchelli_, Mar 17 2024
%e A051634 11 belongs to the sequence because 11 > (7 + 13)/2.
%p A051634 q:= n-> isprime(n) and 2*n>prevprime(n)+nextprime(n):
%p A051634 select(q, [$3..1000])[];  # _Alois P. Heinz_, Jun 21 2023
%t A051634 Transpose[Select[Partition[Prime[Range[10^2]], 3, 1], #[[2]]>(#[[1]]+#[[3]])/2 &]][[2]] (* _Vladimir Joseph Stephan Orlovsky_, May 01 2008 *)
%t A051634 p=Prime[Range[200]]; p[[Flatten[1+Position[Sign[Differences[p,2]], -1]]]]
%o A051634 (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
%o A051634 (Haskell)
%o A051634 a051634 n = a051634_list !! (n-1)
%o A051634 a051634_list = f a000040_list where
%o A051634    f (p:qs@(q:r:ps)) = if 2 * q > (p + r) then q : f qs else f qs
%o A051634 -- _Reinhard Zumkeller_, May 09 2013
%o A051634 (Python)
%o A051634 from sympy import nextprime
%o A051634 def aupto(limit):
%o A051634     alst, p, q, r = [], 2, 3, 5
%o A051634     while q <= limit:
%o A051634         if 2*q > p + r: alst.append(q)
%o A051634         p, q, r = q, r, nextprime(r)
%o A051634     return alst
%o A051634 print(aupto(701)) # _Michael S. Branicky_, Nov 17 2021
%Y A051634 Cf. A006562, A051635, A229832.
%Y A051634 Subsequence of A178943.
%Y A051634 Cf. A225493 (multiplicative closure), A131499 (subsequence).
%K A051634 nice,nonn
%O A051634 1,1
%A A051634 _Felice Russo_, Nov 15 1999

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