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Let f(p) = greatest prime divisor of p-1. Sequence gives smallest prime which takes at least n steps to reach 2 when f is iterated.
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%I #49 Apr 29 2024 21:04:09

%S 2,3,7,23,47,283,719,1439,2879,34549,138197,1266767,14619833,36449279,

%T 377982107,1432349099,22111003847

%N Let f(p) = greatest prime divisor of p-1. Sequence gives smallest prime which takes at least n steps to reach 2 when f is iterated.

%C There is a remarkable and unexplained agreement: if 3 and 7 are replaced by 11 and 14619833 is replaced by 14920303, the result is sequence A056637 (least prime of class n-, according to the Erdős-Selfridge classification of primes).

%C From _David A. Corneth_, Oct 18 2016: (Start):

%C If a(n) * k + 1 is prime then a(n + 1) <= a(n) * k + 1.

%C a(18), a(19), ..., a(23) <= 309554053859, 619108107719, 19811459447009, 433142367554861, 866284735109723, 22523403112852799 respectively. (End)

%C Conjecture: a(n) is the smallest prime p such that b(p) = n, where f(2) = 0 and for an odd prime p, f(p) = 1 + max{q|(p-1), q prime} f(q). In other words, a(n) is the smallest prime p such that A364332(primepi(p)) = n. Verified for n <= 13. - _Jianing Song_, Apr 28 2024

%D Steven G. Johnson, Postings to Number Theory List, Apr 23 and Apr 25, 2003.

%e a(2) = 7 since 7 -> 3 -> 2 takes two steps, and smaller primes require less than 2 steps.

%e For p = 2879, 8 steps are needed (2879 -> 1439 -> 719 -> 359 -> 179 -> 89 -> 11 -> 5 -> 2), so a(8) = 2879, since smaller primes require less than 8 steps.

%t (* Assuming a(n) > 2 a(n-1) if n>1 *) Clear[a, f]; f[p_] := FactorInteger[p - 1][[-1, 1]]; f[2] = 2; a[n_] := a[n] = For[p = NextPrime[2 a[n-1]], True, p = NextPrime[p], k = 0; If[Length[FixedPointList[f, p]] == n+2, Return[p]]]; a[0]=2; a[1]=3; Table[Print[a[n]]; a[n], {n, 0, 16}] (* _Jean-François Alcover_, Oct 18 2016 *)

%Y Cf. A006530, A023503, A083647, A056637, A364332.

%K nonn,more

%O 0,1

%A _N. J. A. Sloane_, Apr 25 2003

%E Edited by _Klaus Brockhaus_, May 01 2003

%E a(16) from _Donovan Johnson_, Nov 17 2008