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A111056 Minimal set of prime-strings in base 10 for primes of the form 4n+3 in the sense of A071062.

%I A111056 #40 Feb 17 2025 12:00:20
%S A111056 3,7,11,19,59,251,491,499,691,991,2099,2699,2999,4051,4451,4651,5051,
%T A111056 5651,5851,6299,6451,6551,6899,8291,8699,8951,8999,9551,9851,22091,
%U A111056 22291,66851,80051,80651,84551,85451,86851,88651,92899,98299,98899
%N A111056 Minimal set of prime-strings in base 10 for primes of the form 4n+3 in the sense of A071062.
%C A111056 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 A111056 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 A111056 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 A111056 Walter A. Kehowski, <a href="/A111055/a111055.txt">Full list of terms</a>
%H A111056 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 A111056 Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_178.htm">Puzzle 178. Shallit Minimal Primes Set</a>, The Prime Puzzles & Problems Connection.
%H A111056 Jeffrey Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf">Minimal primes</a>, J. Recreational Mathematics, vol. 30.2, pp. 113-117, 1999-2000.
%e A111056 From _Danny Rorabaugh_, Mar 26 2015: (Start)
%e A111056 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 A111056 This sequence contains 2099 since 2, 0, 9, 20, 09, 99, 209, 299, and 099 are not primes of the form 4n+3.
%e A111056 (End)
%p A111056 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 A111056 Cf. A071062, A071070, A110600, A110615.
%K A111056 base,fini,nonn,uned
%O A111056 1,1
%A A111056 _Walter Kehowski_, Oct 06 2005