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

5, 13, 17, 29, 37, 41, 61, 73, 89, 97, 101, 109, 149, 181, 233, 277, 281, 349, 409, 433, 449, 677, 701, 709, 769, 821, 877, 881, 1669, 2221, 3001, 3121, 3169, 3221, 3301, 3833, 4969, 4993, 6469, 6833, 6949, 7121, 7477, 7949, 9001, 9049, 9221, 9649, 9833

Offset: 1

Author

Walter Kehowski, Oct 06 2005

Keywords

• base
• fini
• nonn
• full

Comments

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.

The basic algorithm is: if no substring of p matches any previously found prime, add p to the list.

The basic theorem of minimal sets says that the minimal set is always finite.

Examples

			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.
		

Links

• Walter A. Kehowski and Curtis Bright, Table of n, a(n) for n = 1..146 (first 135 terms from Walter A. Kehowski)
• Carlos Rivera, Puzzle 178. Shallit Minimal Primes Set, The Prime Puzzles & Problems Connection.
• Jeffrey Shallit, Minimal primes, J. Recreational Mathematics, vol. 30.2, pp. 113-117, 1999-2000.

Crossrefs

Cf. A071062, A071070, A110600, A110615.

Programs

Maple

with(StringTools);
wc := proc(s) cat("*",Join(convert(s,list),"*"),"*") end;
M1:=[]: wcM1:=[]: 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 = 1 then sp:=convert(p,string); if andmap(proc(w) not(WildcardMatch(w,sp)) end, wcM1) then
M1:=[op(M1),p]; wcM1:=[op(wcM1), wc(sp)]; print(p) fi fi od od;

Extensions

Shortened definition; moved some material from the examples to the comments - R. J. Mathar, May 24 2010