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A181922
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a(0)=5, a(1)=17, and a(n) = least prime p such that p > a(n-1) * log(p) * log(log(p)).
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3
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5, 17, 137, 2141, 55987, 2191523, 119117233, 8524439041, 772233895679, 85935374340821, 11472892288974913, 1802994904127155369, 328370182285306077149, 68412255688529875841713, 16126185570212623152792407, 4260622333455392391536790721
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OFFSET
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0,1
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COMMENTS
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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
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LINKS
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FORMULA
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MATHEMATICA
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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 *)
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PROG
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(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))))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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