A181922 - OEIS

%I #28 Sep 10 2018 15:12:17

%S 5,17,137,2141,55987,2191523,119117233,8524439041,772233895679,

%T 85935374340821,11472892288974913,1802994904127155369,

%U 328370182285306077149,68412255688529875841713,16126185570212623152792407,4260622333455392391536790721

%N a(0)=5, a(1)=17, and a(n) = least prime p such that p > a(n-1) * log(p) * log(log(p)).

%C Broadhurst calls EPPP, for efficient portmanteau probable primes, the terms a(N) such that for all n <= N, a(n-1) = floor(a(n)/d(a(n))), where d(x)=log(x)*log(log(x)). In this case, the prime a(N) efficiently encodes the whole list of the first N terms of this sequence, via this simple "unpacking" formula.

%C It turns out that the terms are EPPP at least up to N=1000. In this range, the gap delta(n)=a(n)/d(a(n))-a(n-1) takes a maximum of ~ 0.94253 at n=27. For example, the 4407 digit PRP a(1000) "contains" the primes a(0), ..., a(1000) with a total of roughly 2 million digits. (As the name indicates, terms listed in the b-file are only PRP.)

%C Since all terms are odd, one could still consider the encoding "safe" as long as delta(n) < 2, i.e., use a(n-1) = floor[a(n)/d(a(n))/2]*2+1. Beyond, one might introduce k-EPPP for an encoding that allows recovery of the preceding terms modulo testing divisibility by p <= k. (This, however, cannot ensure a "safe" encoding as soon as delta >= 2: One might well run into the case where a(n-1) is the lesser if a twin prime pair.) - _M. F. Hasler_, Apr 03 2012

%H D. Broadhurst, <a href="/A181922/b181922.txt">Table of n, a(n) for n = 0..100</a> (longer version given below)

%H D. Broadhurst, <a href="http://physics.open.ac.uk/~dbroadhu/cert/b181922.txt">Table of n, a(n), for n=0,...,1000</a>.

%H D. Broadhurst, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;c6a3edbb.1204">Efficient portmanteau probable primes</a>, Apr 01 2012

%F a(n+1) = A181943(a(n)) and a(n-1) = A181942(a(n)) for all n>0. - _M. F. Hasler_, Apr 04 2012

%t Prepend[NestList[Block[{p = NextPrime@ #}, While[p <= # Log[p] Log@ Log@ p, p = NextPrime@ p]; p] &@ # &, 17, 4], 5] (* _Michael De Vlieger_, Jan 03 2016 *)

%o (PARI) {my(d(x)=log(x)*log(log(x))); print1(5","p=17); for(n=2,20,print1(","p=nextprime(solve(X=p,p^2,X/d(X)-p))))}

%K nonn

%O 0,1

%A _M. F. Hasler_, Apr 02 2012