OFFSET

0,1

COMMENTS

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.

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.)

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

LINKS

D. Broadhurst, Table of n, a(n) for n = 0..100 (longer version given below)

D. Broadhurst, Table of n, a(n), for n=0,...,1000.

D. Broadhurst, Efficient portmanteau probable primes, Apr 01 2012

FORMULA

MATHEMATICA

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 *)

PROG

(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))))}

CROSSREFS

KEYWORD

nonn

AUTHOR

M. F. Hasler, Apr 02 2012

STATUS

approved