# [seqfan] Re: Primes by rank

Robert G. Wilson v rgwv at rgwv.com
Sat May 29 00:10:58 CEST 2010

```Et al,

This this not a 'combinational' approach to A005113 & A056637?

Bob.

--------------------------------------------------
From: "N. J. A. Sloane" <njas at research.att.com>
Sent: Friday, May 28, 2010 12:30 PM
To: <seqfan at seqfan.eu>
Cc: <njas at research.att.com>
Subject: [seqfan]  Primes by rank

>
> Dear Seq Fans, I've edited this entry to clarify the text.
> My question is, what is the sequence   a(n) = rank of n-th prime  which
> underlies this?  For which this sequence gives the records?
> Neil
>
> %I A177854
> %S A177854 2,3,11,131,1571,43717,5032843,1047774137
> %N A177854 Smallest prime of rank n.
> %C A177854 The Brillhart-Lehmer-Selfridge algorithm provides a general
> method for proving the primality of P as long as one can factor P+1 or
> P-1. Therefore for any prime number, when P+1 or P-1 is completely
> factored, the primality of any factors of P+1 or P-1 can also be proved by
> the same algorithm. The longest recursive primality proving chain depth is
> called the rank of P.
> %H A177854 L. Zhou, <a
> href="http://bitc.bme.emory.edu/~lzhou/blogs/?p=117">The rank of
> primes</a>
> %e A177854 The "trivial" prime 2 has rank 0. 3 = 2+1 takes one step to
> reduce to 2, so 3 has rank 1.
> %e A177854 P=131: P+1=132=2^2*3*11. P1[1]=2 has rank 0; P1[2]=3 has rank
> 1; P1[3]=11: P1[3]+1=12=2^2*3; is one step from 3 and has recursion depth
> = 2. So P=131 has total maximum recursion depth 2+1 = 3 and therefore has
> rank 3.
> %t A177854 The following program runs through all prime numbers until it
> finds the first rank 7 prime. (It took about a week.) Fr[n_]:= Module[{nm,
> np, fm, fp, szm, szp, maxm, maxp, thism, thisp, res, jm, jp}, If[n == 2,
> res = 0, nm = n - 1; np = n + 1; fm = FactorInteger[nm]; fp =
> FactorInteger[np]; szm = Length[fm]; szp = Length[fp]; maxm = 0; Do[thism
> = Fr[fm[[jm]][[1]]]; If[maxm < thism, maxm = thism], {jm, 1, szm}]; maxp =
> 0; Do[thisp = Fr[fp[[jp]][[1]]]; If[maxp maxp, res = maxp]; res++ ]; res];
> i=1;While[p = Prime[i]; s = Fr[p];[p, s] >>> "prime_rank.out";s<7,i++ ]
> %K A177854 hard,nonn,more,nice,new
> %O A177854 0,1
> %A A177854 Lei Zhou (lzhou5(AT)emory.edu), May 14 2010
> %E A177854 Partially edited by N. J. A. Sloane, May 15 2010, May 28 2010
>
>
>
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