

A177854


Smallest prime of rank n.


2




OFFSET

0,1


COMMENTS

The BrillhartLehmerSelfridge algorithm provides a general method for proving the primality of P as long as one can factor P+1 or P1. Therefore for any prime number, when P+1 or P1 is completely factored, the primality of any factors of P+1 or P1 can also be proved by the same algorithm. The shortest recursive primality proving chain depth is called the rank of P (cf. A169818).


LINKS

Table of n, a(n) for n=0..7.
L. Zhou, The rank of primes
J. Brillhart, D. H. Lehmer and J. L. Selfridge, New primality criteria and factorizations of 2^m+1, Math. Compl. 29 (1975) 620647.
Wikipedia, LucasLehmerRiesel test.


EXAMPLE

The "trivial" prime 2 has rank 0. 3 = 2+1 takes one step to reduce to 2, so 3 has rank 1.
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.


MATHEMATICA

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++ ]


PROG

(PARI) rank(p)=if(p<8, return(p>2)); vecmin(apply(k>vecmax(apply(rank, factor(k)[, 1])), [p1, p+1]))+1
print1(2); r=0; forprime(p=3, , t=rank(p); if(t>r, r=t; print1(", "p))) \\ Charles R Greathouse IV, Oct 03 2016


CROSSREFS

These are the primes where records occur in A169818.
Cf. A005113, A056637.  Robert G. Wilson v, May 28 2010
Sequence in context: A058114 A042337 A061482 * A273598 A135161 A066100
Adjacent sequences: A177851 A177852 A177853 * A177855 A177856 A177857


KEYWORD

hard,nonn,more,nice


AUTHOR

Lei Zhou, May 14 2010


EXTENSIONS

Partially edited by N. J. A. Sloane, May 15 2010, May 28 2010
Definition corrected by Robert Gerbicz, May 28 2010


STATUS

approved



