%I #26 Nov 04 2024 01:35:58

%S 2,7,1129,1327,19609,31397,155921,370261,1357201,2010733,20831323,

%T 20831323,191912783,436273009,3842610773,10726904659,25056082087,

%U 25056082087,25056082087,1346294310749,1408695493609,2614941710599,13829048559701,19581334192423,19581334192423

%N Smallest prime p such that q = (r-p)/log(p) > n, where r is the next prime after p.

%C Is lim superior(q(n)) = +infinity? See A082892.

%H Amiram Eldar, <a href="/A082891/b082891.txt">Table of n, a(n) for n = 1..32</a> (calculated using the data at A111870)

%F a(n)= Min{p(x); (p(x+1)-p(x))/log(p(x)) > n}.

%e For n = 11 and 12: k = 1319945: p(k+1) = 20831533, p(k) = 20831323, d = p(k+1) - p(k) = 210, log(20831321) = 16.852..., q = 210/16.852... = 12.4615... > 12 and also > 11 for the first time, so a(11) = a(12) = 20831323.

%t Do[s=(Prime[n+1]-Prime[n])/Log[Prime[n]]//N; If[s>11, Print[{n, Prime[n], Prime[n+1], s, Log[Prime[n]]//N}]], {n, 1000000, 100000000}]

%o (PARI) lista(pmax) = {my(n = 1, prv = 2, d, m); print1(2, ", "); forprime(p=3, pmax, d = p-prv; m = floor(d/log(prv)); if(m > n, for(k = 1, m-n, print1(prv, ", ")); n = m); prv=p);} \\ _Amiram Eldar_, Nov 04 2024

%Y Cf. A082862, A082884, A082885, A082886, A082888, A082889, A082890, A082892, A111870.

%K nonn,changed

%O 1,1

%A _Labos Elemer_, Apr 17 2003

%E a(10) corrected and a(13)-a(25) added by _Amiram Eldar_, Nov 04 2024