A110600 -id:A110600 - cp's OEIS frontend

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

Walter Kehowski, Oct 06 2005

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.

			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.

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.

Cf. A071062, A071070, A110600, A110615.

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;

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

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

Original entry on oeis.org

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

Walter Kehowski, Oct 06 2005

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

			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)

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.

Cf. A071062, A071070, A110600, A110615.

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.

A326609 Largest minimal prime in base n (written in base 10).

Original entry on oeis.org

3, 13, 5, 3121, 5209, 2801, 76695841, 811, 66600049, 29156193474041220857161146715104735751776055777, 388177921

Offset: 2

Richard N. Smith, Jul 13 2019

a(13) is (probably) 13^32020*8+183, it has 35670 digits, a(14) = 14^85*4+65, it has 99 digits, a(15) = (15^106*66-619)/7, it has 126 digits, a(16) = 16^3544*9+145, it has 4269 digits.
a(17) is the smallest prime of the form (4105*17^k-9)/16 if it exists, otherwise (probably) (73*17^111333-9)/16 (136991 digits), a(18) = 18^31*304+1 (42 digits).
Other known terms: a(20) = (20^449*16-2809)/19 (585 digits), a(22) = 22^763*20+7041 (1026 digits), a(23) is (probably) (23^800873*106-7)/11 (1090573 digits), a(24) = (24^99*512-121)/23 (138 digits), a(30) = 30^1023*12+1 (1513 digits), a(42) = (42^487*27-1093)/41 (791 digits).
a(19) is the smallest prime of the form (15964*19^k-1)/3 if it exists, otherwise (probably) (904*19^110984-1)/3 (141924 digits), a(21) is the smallest prime of the form 16*21^k+335 if it exists, otherwise (probably) (51*21^479149-1243)/4 (633542 digits).

Richard N. Smith, Table of n, a(n) for n = 2..16
Curtis Bright, Raymond Devillers, and Jeffrey Shallit, Minimal Elements for the Prime Numbers, preprint, 2014.
Curtis Bright, Raymond Devillers, and Jeffrey Shallit, Minimal Elements for the Prime Numbers, Experimental Mathematics, Volume 25, 2016 - Issue 3.
C. K. Caldwell, The Prime Glossary, minimal prime
Richard N. Smith, List of all known terms
Top PRPs, Search for 8*13^n+183
Top PRPs, Search for (73*17^n-9)/16
Top PRPs, Search for (106*23^n-7)/11
Wikipedia, Minimal prime (recreational mathematics)

Cf. A071062 (base 10 minimal primes), A110600 (base 12 minimal primes).

Cf. A293142 (largest non-repunit permutable prime), A317689 (largest non-repunit circular prime), A103443 (largest left-truncatable prime), A023107 (largest right-truncatable prime), A323137 (largest two-sided prime), A084738 (smallest repunit prime), A186995 (smallest weakly prime).

A111057 Minimal set in the sense of A071062 of prime-strings in base 12 for primes of the form 4n+1.

Original entry on oeis.org

5, 13, 37, 73, 97, 109, 313, 337, 373, 409, 421, 577, 601, 661, 709, 1009, 1033, 1093, 1129, 1489, 1609, 1669, 3457, 7537, 12721, 13729, 17401, 17569, 19009, 19141, 20593, 20641, 165877, 208501, 221173, 225781, 226201, 226357, 228793, 246817, 246937, 248821, 1097113, 2695813, 2735269, 2736997, 2737129, 32555521, 388177921

Offset: 1

Walter Kehowski, Oct 06 2005

Maple worksheet available upon request. Here is the minimal set of primes of the form 4n+1 in base 12, where X is ten and E is eleven. 5, 11, 31, 61, 81, 91, 221, 241, 271, 2X1, 2E1, 401, 421, 471, 4E1, 701, 721, 771, 7X1, X41, E21, E71, 2001, 4441, 7441, 7E41, X0X1, X201, E001, E0E1, EE01, EE41, 7EEE1, X07E1, X7EE1, XX7E1, XXXX1, XXEE1, E04X1, EXX01, EXXX1, EEEE1, 44XXX1, XX00E1, XEXXE1, XEEXE1, XEEEX1, XXX0001, XX000001. Note that the last prime in the set is the same as the last prime in the minimal set of all primes. See A110600. I am checking certain ranges past this last prime but flow-charting the possibilities leads me to believe I have found the full sequence. The minimal set of prime strings in base 12 for primes of the form 4n+3 is [3, 7, E] since every 4n+3 prime greater than 3 ends in either 7 or E.

			a(11)=421="2E1" since the pattern "*2*E*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 "401" (577 in base 10) is the next prime in the list. The basic rule is: if no substring of p matches any previously found prime, add p to the list. The basic theorem of minimal sets says that this process will terminate, that is, the minimal set is always finite.

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

Cf. A071062, A071070, A110600, A110615.

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

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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.

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A326609 Largest minimal prime in base n (written in base 10).

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A111057 Minimal set in the sense of A071062 of prime-strings in base 12 for primes of the form 4n+1.

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