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A246781 Numbers n such that A182134(n) = 3, i.e., there exist only three primes p with prime(n) < p < prime(n)^(1 + 1/n).

%I A246781 #35 Mar 23 2015 18:51:15
%S A246781 12,13,16,18,20,21,27,31,34,39,44,53,59,60,65,96,97,98,99,136,154,202,
%T A246781 214,215,220,221,280,324,325,326,365,366,736,780,2146,2225,3792,5946,
%U A246781 5947,5948,6902,6903,18524,22078,23510,23511,23512,31542,31544,33606
%N A246781 Numbers n such that A182134(n) = 3, i.e., there exist only three primes p with prime(n) < p < prime(n)^(1 + 1/n).
%C A246781 Firoozbakht's conjecture states that for every n, there exists at least one prime p with prime(n) < p < prime(n)^(1+1/n).
%C A246781 The only known indices n for which A182134(n) = 1 are {1, 2, 3, 4, 8}.
%C A246781 See A246782 for indices n such that A182134(n) = 2.
%C A246781 This sequence lists numbers n such that A182134(n) = 3.
%H A246781 Robert Price, <a href="/A246781/b246781.txt">Table of n, a(n) for n = 1..170</a>
%H A246781 Carlos Rivera, <a href="http://www.primepuzzles.net/conjectures/conj_030.htm">Conjecture 30. The Firoozbakht Conjecture</a> PrimePuzzles.net.
%H A246781 A. Kourbatov, <a href="http://arxiv.org/abs/1503.01744">Verification of the Firoozbakht conjecture for primes up to four quintillion</a>, arXiv:1503.01744 [math.NT], 2015
%H A246781 Wikipedia, <a href="http://en.wikipedia.org/wiki/Firoozbakht%E2%80%99s_conjecture">Firoozbakht's conjecture</a>
%e A246781 12 is in the sequence since there exists only three primes p where, prime(12) < p < prime(12)^(1 + 1/12). Note that prime(12) = 37, 37^(1 + 1/12) ~ 49.99 and 37 < 41 < 43 < 47 < 49.99.
%p A246781 N:= 10^5: # to get all terms where prime(n)^(1+1/n) < N
%p A246781 Primes:= select(isprime,[2,seq(2*i+1,i=1..floor((N+1)/2))]):
%p A246781 filter:= proc(n) local t; t:= Primes[n]^(n+1); Primes[n+3]^n <= t and Primes[n+4]^n > t end proc:
%p A246781 select(filter, [$1..nops(Primes)-4]); # _Robert Israel_, Mar 23 2015
%t A246781 np[n_] := (a = Prime[n]; b = a^(1 + 1/n); Length[Select[Range[a + 1, b], PrimeQ]]); Select[Range[10000], np[#] == 3 &]
%o A246781 (Haskell)
%o A246781 a246781 n = a246781_list !! (n-1)
%o A246781 a246781_list = filter ((== 3) . a182134) [1..]
%o A246781 -- _Reinhard Zumkeller_, Nov 17 2014
%Y A246781 Cf. A000040, A182134, A246782.
%Y A246781 Cf. A249566.
%K A246781 nonn
%O A246781 1,1
%A A246781 _Farideh Firoozbakht_, Oct 12 2014
%E A246781 a(43)-a(50) from _Robert Price_, Oct 24 2014