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

1,1

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.

Links

Walter A. Kehowski and Curtis Bright, Table of n, a(n) for n = 1..146 (first 135 terms from Walter A. Kehowski)

C. Rivera, Shallit Minimal Primes Set (Puzzle No. 178), PrimePuzzle.net.

J. Shallit, Minimal primes, J. Recreational Mathematics, vol. 30.2, pp. 113-117, 1999-2000.

Example

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.

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;

Crossrefs

Cf. A071062, A071070, A110600, A110615.

Sequence in context: A357218 A349900 A347163 * A307096 A283391 A145016

Adjacent sequences: A111052 A111053 A111054 * A111056 A111057 A111058

Keyword

base,fini,nonn,full

Author

Walter Kehowski, Oct 06 2005

Extensions

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

Status

approved