|
%I #37 May 15 2015 08:30:08
%S 3,7,11,19,59,251,491,499,691,991,2099,2699,2999,4051,4451,4651,5051,
%T 5651,5851,6299,6451,6551,6899,8291,8699,8951,8999,9551,9851,22091,
%U 22291,66851,80051,80651,84551,85451,86851,88651,92899,98299,98899
%N Minimal set of prime-strings in base 10 for primes of the form 4n+3 in the sense of A071062.
%C The basic rule is: if no substring of p matches any smaller prime of the form 4n+3, add p to the list. The basic theorem of minimal sets says that the minimal set is always finite.
%C The sequence b-file is complete except for the number (2*10^19153 + 691)/9, i.e., the decimal number consisting of 19151 "2"s followed by two "9"s. - _Curtis Bright_, Jan 23 2015
%H Walter A. Kehowski and Curtis Bright, <a href="/A111056/b111056.txt">Table of n, a(n) for n = 1..112</a> (first 103 terms from Walter A. Kehowski)
%H Walter A. Kehowski, <a href="/A111055/a111055.txt">Full list of terms</a>
%H C. Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_178.htm">Shallit Minimal Primes Set (Puzzle No. 178)</a>, PrimePuzzle.net.
%H F. Morain, <a href="http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html">Primality certificate</a> for the largest number of A111056, May 4 2015
%H J. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.ps">Minimal primes</a>, J. Recreational Mathematics, vol. 30.2, pp. 113-117, 1999-2000.
%e From _Danny Rorabaugh_, Mar 26 2015: (Start)
%e a(5) is not 23, even though 23 is the fifth prime of the form 4n+3, since 23 contains a(1)=3 as a substring. Similarly: 31 and 43 contain 3 and 47 contains a(2)=7. Thus a(5)=59.
%e This sequence contains 2099 since 2, 0, 9, 20, 09, 99, 209, 299, and 099 are not primes of the form 4n+3.
%e (End)
%p with(StringTools); wc := proc(s) cat("*",Join(convert(s,list),"*"),"*") end; M3:=[]: wcM3:=[]: p:=1: for z from 1 to 1 do for k while p<10^11 do p:=nextprime(p); if k mod 100000 = 0 then print(k,p,evalf((time()-st)/60,4)) fi; if p mod 4 = 3 then sp:=convert(p,string); if andmap(proc(w) not(WildcardMatch(w,sp)) end, wcM3) then M3:=[op(M3),p]; wcM3:=[op(wcM3),wc(sp)]; print(p) fi fi od od; # Let it run for a couple of days.
%Y Cf. A071062, A071070, A110600, A110615.
%K base,fini,nonn,uned
%O 1,1
%A _Walter Kehowski_, Oct 06 2005
|
