%I #43 Mar 04 2016 04:35:31

%S 5,6,7,9,10,11,14,15,22,23,28,29,30,45,46,61,66,216,217,367,3793,

%T 1319945,1576499,8040877,17567976,44405858,445538764,1478061204,

%U 3643075047,17440041685,190836014732,714573709895,714573709896

%N Numbers n such that A182134(n)=2, i.e., there exist only two primes p with prime(n) < p < prime(n)^(1+1/n).

%C Firoozbakht's conjecture says that for every n, there exists at least one prime p such that prime(n) < p < prime(n)^(1+1/n).

%C Let A(m) = {n | A182134(n) = m} where A182134(n) = #{p | p is prime and prime(n) < p < prime(n)^(1+1/n)}. This sequence gives the terms of A(2) and the sequence A246781 gives the terms of A(3).

%C The only known indices n for which A182134(n) = 1 are {1, 2, 3, 4, 8}. It is conjectured that this is the complete set A(1).

%C Conjecture: For all m, where m is greater than one, A(m) is an infinite set.

%C a1 = 49749629143524, a2 = 1475067052906944 and a3 = 1475067052906945 are three large terms of the sequence. It is interesting that a3 - a2 = 1.

%C Conjecture: The sequence is infinite.

%C Next term is greater than 25000000.

%C a(34) > 10^12. - _Robert Price_, Nov 01 2014

%C The conjecture that A(1)={1, 2, 3, 4, 8} holds through 10^12. - _Robert Price_, Nov 01 2014

%H 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 Wikipedia, <a href="http://en.wikipedia.org/wiki/Firoozbakht%E2%80%99s_conjecture">Firoozbakht's conjecture</a>

%e 5 is in the sequence since there exists only two primes p, prime(5) < p < prime(5)^(1+1/5). Note that prime(5) = 11, 11^(1+1/5)) ~ 17.77 and 11 < 13 < 17 < 17.77.

%t np[n_]:=(a = Prime[n]; b = a^(1 + 1/n); Length[Select[Range[a+1,b], PrimeQ]]); Do[If[np[n] == 2,Print[n]], {n, 25000000}]

%o (PARI) for(n=1,9e9,2==primepi(prime(n)^(1+1/n))-n&&print1(n",") \\ _M. F. Hasler_, Nov 03 2014

%o (Haskell)

%o a246782 n = a246782_list !! (n-1)

%o a246782_list = filter ((== 2) . a182134) [1..]

%o -- _Reinhard Zumkeller_, Nov 17 2014

%Y Cf. A000040, A002386, A005669, A182134, A246781, A249566.

%K nonn,more

%O 1,1

%A _Farideh Firoozbakht_, Oct 12 2014

%E a(26)-a(27) from _Robert Price_, Oct 24 2014

%E a(28)-a(33) from _Robert Price_, Nov 01 2014