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

3, 7, 11, 19, 59, 251, 491, 499, 691, 991, 2099, 2699, 2999, 4051, 4451, 4651, 5051, 5651, 5851, 6299, 6451, 6551, 6899, 8291, 8699, 8951, 8999, 9551, 9851, 22091, 22291, 66851, 80051, 80651, 84551, 85451, 86851, 88651, 92899, 98299, 98899

Offset: 1

Author

Walter Kehowski, Oct 06 2005

Keywords

• base
• fini
• nonn
• uned

Comments

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.

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

Examples

			From _Danny Rorabaugh_, Mar 26 2015: (Start)
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.
This sequence contains 2099 since 2, 0, 9, 20, 09, 99, 209, 299, and 099 are not primes of the form 4n+3.
(End)
		

Links

• Walter A. Kehowski and Curtis Bright, Table of n, a(n) for n = 1..112 (first 103 terms from Walter A. Kehowski)
• Walter A. Kehowski, Full list of terms
• F. Morain, Primality certificate for the largest number of A111056, May 4 2015.
• 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; 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.