cp's OEIS Frontend

A082891 Smallest prime p such that q = (r-p)/log(p) > n, where r is the next prime after p.

%I A082891 #26 Nov 04 2024 01:35:58
%S A082891 2,7,1129,1327,19609,31397,155921,370261,1357201,2010733,20831323,
%T A082891 20831323,191912783,436273009,3842610773,10726904659,25056082087,
%U A082891 25056082087,25056082087,1346294310749,1408695493609,2614941710599,13829048559701,19581334192423,19581334192423
%N A082891 Smallest prime p such that q = (r-p)/log(p) > n, where r is the next prime after p.
%C A082891 Is lim superior(q(n)) = +infinity? See A082892.
%H A082891 Amiram Eldar, <a href="/A082891/b082891.txt">Table of n, a(n) for n = 1..32</a> (calculated using the data at A111870)
%F A082891 a(n)= Min{p(x); (p(x+1)-p(x))/log(p(x)) > n}.
%e A082891 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 A082891 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 A082891 (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 A082891 Cf. A082862, A082884, A082885, A082886, A082888, A082889, A082890, A082892, A111870.
%K A082891 nonn
%O A082891 1,1
%A A082891 _Labos Elemer_, Apr 17 2003
%E A082891 a(10) corrected and a(13)-a(25) added by _Amiram Eldar_, Nov 04 2024