A082891 Smallest prime p such that q = (r-p)/log(p) > n, where r is the next prime after p.
2, 7, 1129, 1327, 19609, 31397, 155921, 370261, 1357201, 2010733, 20831323, 20831323, 191912783, 436273009, 3842610773, 10726904659, 25056082087, 25056082087, 25056082087, 1346294310749, 1408695493609, 2614941710599, 13829048559701, 19581334192423, 19581334192423
Offset: 1
Keywords
Examples
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.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..32 (calculated using the data at A111870)
Programs
-
Mathematica
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}] -
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
Formula
a(n)= Min{p(x); (p(x+1)-p(x))/log(p(x)) > n}.
Extensions
a(10) corrected and a(13)-a(25) added by Amiram Eldar, Nov 04 2024
Comments