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A111055 The set of primes of the form 4n+1 that is minimal in the sense of A071062.

%I A111055 #24 Feb 17 2025 12:00:14
%S A111055 5,13,17,29,37,41,61,73,89,97,101,109,149,181,233,277,281,349,409,433,
%T A111055 449,677,701,709,769,821,877,881,1669,2221,3001,3121,3169,3221,3301,
%U A111055 3833,4969,4993,6469,6833,6949,7121,7477,7949,9001,9049,9221,9649,9833
%N A111055 The set of primes of the form 4n+1 that is minimal in the sense of A071062.
%C A111055 This means: by removing any (possibly none) of the decimal digits of any member of A002144 one can obtain some number of this sequence here.
%C A111055 The basic algorithm is: if no substring of p matches any previously found prime, add p to the list.
%C A111055 The basic theorem of minimal sets says that the minimal set is always finite.
%H A111055 Walter A. Kehowski and Curtis Bright, <a href="/A111055/b111055.txt">Table of n, a(n) for n = 1..146</a> (first 135 terms from Walter A. Kehowski)
%H A111055 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 A111055 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 A111055 a(11)=101 since the pattern "*1*0*1*" does not occur in any previously found prime of the form 4n+1. Assuming all previous members of the list have been similarly recursively constructed, then 109 is the next prime in the list.
%p A111055 with(StringTools);
%p A111055 wc := proc(s) cat("*",Join(convert(s,list),"*"),"*") end;
%p A111055 M1:=[]: wcM1:=[]: p:=1: for z from 1 to 1 do for k while p<10^11 do p:=nextprime(p);
%p A111055 if k mod 100000 = 0 then print(k,p,evalf((time()-st)/60,4)) fi;
%p A111055 if p mod 4 = 1 then sp:=convert(p,string); if andmap(proc(w) not(WildcardMatch(w,sp)) end, wcM1) then
%p A111055 M1:=[op(M1),p]; wcM1:=[op(wcM1), wc(sp)]; print(p) fi fi od od;
%Y A111055 Cf. A071062, A071070, A110600, A110615.
%K A111055 base,fini,nonn,full
%O A111055 1,1
%A A111055 _Walter Kehowski_, Oct 06 2005
%E A111055 Shortened definition; moved some material from the examples to the comments - _R. J. Mathar_, May 24 2010