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%I #72 Sep 08 2022 08:46:09
%S 2,2,3,4,6,6,8,8,9,11,11,12,13,13,13,14,15,15,16,16,16,17,17,18,19,19,
%T 19,19,19,19,21,21,22,21,22,22,22,23,23,23,24,23,24,24,24,24,25,26,26,
%U 26,26,26,26,27,27,27,27,27,27,27,27,28,29,29,28,28,29,30,30,30
%N a(n) = floor(prime(n)^(1+1/n)) - prime(n).
%C The Firoozbakht Conjecture, "prime(n)^(1/n) is a strictly decreasing function of n" is true if and only if a(n) - A001223(n) is nonnegative for all n. The conjecture is true for all primes p where p < 4.0*10^18. (See A. Kourbatov link.)
%C 0, 1, 5, 7, 10 & 20 are not in the sequence. It seems that these six integers are all the nonnegative integers which are not in the sequence.
%C From _Alexei Kourbatov_, Nov 27 2015: (Start)
%C Theorem: if prime(n+1) - prime(n) < prime(n)^(3/4), then every integer > 20 is in this sequence.
%C Proof: Let f(n) = prime(n)^(1+1/n) - prime(n). Then a(n) = floor(f(n)).
%C Define F(x) = log^2(x) - log(x) - 1. Using the upper and lower bounds for f(n) established in Theorem 5 of J. Integer Sequences Article 15.11.2; arXiv:1506.03042 we have F(prime(n))-3.83/(log prime(n)) < f(n) < F(prime(n)) for n>10^6; so f(n) is unbounded and asymptotically equal to F(prime(n)).
%C Therefore, for every n>10^6, jumps in f(n) are less than F'(x)*x^(3/4)+3.83/(log x) at x=prime(n), which is less than 1 as x >= prime(10^6)=15485863. Thus jumps in a(n) cannot be more than 1 when n>10^6. Separately, we verify by direct computation that a(n) takes every value from 21 to 256 when 30 < n <= 10^6. This completes the proof.
%C (End)
%D Paulo Ribenboim, The little book of bigger primes, second edition, Springer, 2004, p. 185.
%H Robert G. Wilson v, <a href="/A246778/b246778.txt">Table of n, a(n) for n = 1..10000</a>
%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 A. Kourbatov, <a href="http://arxiv.org/abs/1506.03042">Upper bounds for prime gaps related to Firoozbakht's conjecture</a>, arXiv:1506.03042 [math.NT], 2015.
%H A. Kourbatov, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Kourbatov/kourb7.html">Upper bounds for prime gaps related to Firoozbakht's conjecture</a>, Journal of Integer Sequences, 18 (2015), Article 15.11.2.
%H Carlos Rivera, <a href="http://www.primepuzzles.net/conjectures/conj_030.htm">Conjecture 30</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Firoozbakht%E2%80%99s_conjecture">Firoozbakht's conjecture</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_gap">Prime gap</a>.
%F a(n) = A249669(n) - A000040(n). - _M. F. Hasler_, Nov 03 2014
%F a(n) = (log(prime(n)))^2 - log(prime(n)) + O(1), see arXiv:1506.03042. - _Alexei Kourbatov_, Sep 06 2015
%p N:= 10^4: # to get entries corresponding to all primes <= N
%p Primes:= select(isprime, [2,seq(2*i+1,i=1..floor((N-1)/2))]):
%p seq(floor(Primes[n]^(1+1/n) - Primes[n]), n=1..nops(Primes)); # _Robert Israel_, Mar 23 2015
%t f[n_] := Block[{p = Prime@ n}, Floor[p^(1 + 1/n)] - p]; Array[f, 75]
%o (Magma) [Floor(NthPrime(n)^(1+1/n)) - NthPrime(n): n in [1..70]]; // _Vincenzo Librandi_, Mar 24 2015
%o (PARI) first(m)=vector(m,i,floor(prime(i)^(1+1/i)) - prime(i)) \\ _Anders Hellström_, Sep 06 2015
%Y Cf. A000040, A001223, A246776, A246777, A246779, A246780, A249669.
%K nonn
%O 1,1
%A _Farideh Firoozbakht_, Sep 26 2014
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