cp's OEIS Frontend

A262061 Least prime(i) such that prime(i)^(1+1/i) - prime(i) > n.

%I A262061 #29 Feb 22 2018 16:03:27
%S A262061 2,3,5,7,11,11,17,17,23,29,29,37,41,53,59,67,79,89,97,127,127,137,163,
%T A262061 179,211,223,251,293,307,337,373,419,479,521,541,587,691,727,797,853,
%U A262061 929,1009,1151,1201,1277,1399,1523,1693,1777,1931,2053,2203,2333,2521,2647,2953,3119,3299,3527,3847,4127
%N A262061 Least prime(i) such that prime(i)^(1+1/i) - prime(i) > n.
%C A262061 Where A246778(i) first exceeds n, stated by p_i.
%C A262061 Similar to A245396.
%C A262061 Number of terms < 10^n: 4, 19, 41, 75, 120, 176, 242, 319, 407, 506, ..., .
%C A262061 Concerning Firoozbakht's Conjecture (1982): (prime(n+1))^(1/(n+1)) < prime(n)^(1/n), for all n = 1 or prime(n+1) < prime(n)^(1+1/n), which can be rewritten as: (log(prime(n+1))/log(prime(n)))^n < (1+1/n)^n. This suggests a weaker conjecture: (log(prime(n+1))/log(prime(n)))^n < e.
%C A262061 Prime index of a(n): 1, 1, 3, 4, 5, 5, 7, 7, 9, 10, 10, 12, 13, 16, 17, 19, 22, 24, 25, 31, 31, ..., .
%C A262061 All terms are unique for n > 21. Indices not unique: 1 & 2, 5 & 6, 7 & 8, 10 & 11 and 20 & 21.
%C A262061 The distribution of initial digits, 1...9, for a(n), n<508: 140, 91, 60, 50, 44, 36, 32, 27 and 26.
%D A262061 Paulo Ribenboim, The little book Of bigger primes, second edition, Springer, 2004, p. 185.
%H A262061 Farideh Firoozbakht and Robert G. Wilson v, <a href="/A262061/b262061.txt">Table of n, a(n) for n = 1..507</a>
%H A262061 Alexei 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 A262061 Alexei 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
%F A262061 Log(y) ~= g + x^(1/2) where g = Euler's Gamma.
%e A262061 a(20) = 127 since for all primes less than the 31st prime, 127, p_k^(32/31) - p_k are less than 20.
%e A262061 a(100) = 38113,
%e A262061 a(200) = 2400407,
%e A262061 a(300) = 57189007,
%e A262061 a(400) = 828882731,
%e A262061 a(500) = 8748565643,
%e A262061 a(1000) = 91215796479037,
%e A262061 a(1064) = 246842748060263, limit of Mathematica by direct computation, i.e., the first Mathematica line.
%t A262061 f[n_] := Block[{p = 2, k = 1}, While[n > p^(1 + 1/k) - p, p = NextPrime@ p; k++]; p]; Array[f, 60] (* or  quicker *)
%t A262061 (* or quicker *) p = 2; i = 1; lst = {}; Do[ While[ p^(1 + 1/i) < n + p, p = NextPrime@ p; i++]; AppendTo[lst, p]; Print[{n, p}], {n, 100}]; lst
%o A262061 (PARI) a(n) = {i = 0; forprime(p=2,, i++; if (p^(1+1/i) - p > n, return (p)););} \\ _Michel Marcus_, Oct 04 2015
%Y A262061 Cf. A144104, A182134, A182514, A245396, A246777, A246778, A246781, A246810, A248855, A249566.
%K A262061 nonn
%O A262061 1,1
%A A262061 _Farideh Firoozbakht_ and _Robert G. Wilson v_, Sep 11 2015
%E A262061 a(2) corrected in b-file by _Andrew Howroyd_, Feb 22 2018