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A181922 a(0)=5, a(1)=17, and a(n) = least prime p such that p > a(n-1) * log(p) * log(log(p)). 3
5, 17, 137, 2141, 55987, 2191523, 119117233, 8524439041, 772233895679, 85935374340821, 11472892288974913, 1802994904127155369, 328370182285306077149, 68412255688529875841713, 16126185570212623152792407, 4260622333455392391536790721 (list; graph; refs; listen; history; text; internal format)
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

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

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

Sequence in context: A248661 A176133 A071057 * A177509 A160611 A281429

Adjacent sequences:  A181919 A181920 A181921 * A181923 A181924 A181925

KEYWORD

nonn

AUTHOR

M. F. Hasler, Apr 02 2012

STATUS

approved

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Last modified March 28 13:21 EDT 2020. Contains 333089 sequences. (Running on oeis4.)