A071070 -id:A071070 - cp's OEIS frontend

A110615 Minimal set of composite-strings in base 12 in the sense of A071070.

4, 6, 8, 9, 10, 12, 14, 15, 24, 25, 26, 27, 35, 36, 38, 39, 60, 62, 63, 65, 84, 85, 86, 87, 91, 95, 132, 133, 134, 135, 143, 161, 205, 215, 355, 377, 445, 451, 455, 493, 1651, 1673, 1885, 1891, 1895, 8797

Offset: 1

Walter Kehowski, Sep 14 2005; name corrected Sep 18 2005

nonn

Maple worksheet available upon request. Here is the sequence of minimal composites in base 12, where X is 10 and E is 11. 4, 6, 8, 9, X, 10, 12, 13, 20, 21, 22, 23, 2E, 30, 32, 33, 50, 52, 53, 55, 70, 71, 72, 73, 77, 7E, E0, E1, E2, E3, EE, 115, 151, 15E, 257, 275, 311, 317, 31E, 351, E57, E75, 1111, 1117, 111E, 5111.

			a(13)=35=2E since no earlier composite is of the form "*2*E*". The list of minimal composites can be constructed using a sieve-like process: subsequently eliminate from the list of composites all composites of the form "*2*E*". Assuming all previous terms have been similarly determined, then the next remaining composite should be 30.

Cf. A071062, A071070.

A071062 Minimal set of prime-strings in base 10.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049

Offset: 1

Benoit Cloitre, May 26 2002

Any prime number contains in its digits at least one of the terms of this sequence and there is no smaller set. There are 26 terms in the sequence.

Ioulia N. Baoulina, Martin Kreh, and Jörn Steuding, Deleting digits, arXiv:1607.01548 [math.NT], 2016.
David Butler, 65536, Making Your Own Sense blog, June 2017.
C. K. Caldwell, The Prime Glossary, minimal prime
J.-P. Delahaye, Nombres premiers inévitables et pyramidaux, Pour la science, (French edition of Scientific American), Juin 2002, p. 99.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
J. Shallit, Minimal primes, pdf, J. Recreational Mathematics, vol. 30.2, pp. 113-117, 1999-2000.

Cf. A071070, A071071, A071072, A071073.

Typo corrected by T. D. Noe, Nov 15 2006

Typo corrected by Walter Kehowski, Feb 22 2007

A071071 Minimal "powers of 2" set in base 10: any power of 2 contains at least one term of this sequence in its decimal expansion.

Original entry on oeis.org

1, 2, 4, 8, 65536

Offset: 1

Benoit Cloitre, May 26 2002

Conjectured by J. Shallit to be complete.

A possible exception are powers of 16. It can be proved that 16^(5^(k-1) + floor((k+3)/4)) == 16^floor((k+3)/4) (mod 10^k) (see attached proof). Thus it may be that there is a power of 16 that does not contain any of the digits 1, 2, 4, and 8 or the number 65536 as a substring. - Bassam Abdul-Baki, Apr 10 2019

J.-P. Delahaye, Nombres premiers inévitables et pyramidaux, Pour la science, (French edition of Scientific American), Juin 2002, p. 98

Bassam Abdul-Baki, Minimal Sets for Powers of 2
David Butler, 65536, Making Your Own Sense, June 21 2017.
Jeffrey Shallit, The Prime Game, Recursivity, December 01 2006.
Index to divisibility sequences

Cf. A071062, A071070, A071072, A071073.

A071073 Minimal "multiples of 3" set in base 10.

Original entry on oeis.org

0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 42, 45, 48, 51, 54, 57, 72, 75, 78, 81, 84, 87, 111, 114, 117, 141, 144, 147, 171, 174, 177, 222, 225, 228, 252, 255, 258, 282, 285, 288, 411, 414, 417, 441, 444, 447, 471, 474, 477, 522, 525, 528, 552, 555, 558, 582, 585, 588

Offset: 1

Benoit Cloitre, May 26 2002

Any multiple of 3 contains in its digits at least one of the terms of this sequence. There are 76 terms in the sequence; Delahaye gives all 76 terms and proves that there are no further terms (his statement that there are 280 terms seems to be a typo). There is no smaller set.

Nathaniel Johnston, Table of n, a(n) for n = 1..76 (full sequence)
J.-P. Delahaye, Nombres premiers inévitables et pyramidaux, Pour la Science 296 (2002), p 102.

Cf. A071062, A071071, A071072, A071070.

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

Original entry on oeis.org

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

A071072 Minimal "multiples of 4" set in base 10.

Original entry on oeis.org

0, 4, 8, 12, 16, 32, 36, 52, 56, 72, 76, 92, 96

Offset: 1

Benoit Cloitre, May 26 2002

Any multiple of 4 contains in its digits at least one of the terms of this sequence, and there is no smaller set.

J.-P. Delahaye, "Pour la science" (French edition of Scientific American), Juin 2002, p. 98

Cf. A071062, A071071, A071070, A071073.

A110600 Minimal set of prime-strings in base 12 in the sense of A071062.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 73, 97, 109, 577, 1489, 7537, 17401, 226201, 1097113, 32555521, 388177921

Offset: 1

Walter Kehowski, Sep 14 2005

Maple worksheet available upon request. Here is the minimal set in base 12 where X is 10 and E is 11. 2, 3, 5, 7, E, 11, 61, 81, 91, 401, X41, 4441, X0X1, XXXX1, 44XXX1, XXX0001, XX000001. This minimal set demonstrates the elegance of base 12 generally since you can mentally follow the process of elimination, all primes after E end in the neutral digit 1 and the last two entries only contain X, 0 and 1. There are no primes of the form X0...01 since the sum of its digits is E and hence it is divisible by E.

The smallest prime found to date that satisfies all patterns in the minimal set is 1234456789X04XXX00E0001 (656969693573113867991809 in base 10). [Walter Kehowski, May 18 2012]

			a(10)=401 since no earlier prime in the list contained the pattern "*4*0*1*" where "*" stands for zero or more digits. The list can be manually constructed using a sieve-like process: eliminate all subsequent primes of the form "*4*0*1*" from the list of all primes. Assuming all previous elements have also been similarly determined, the next remaining prime should be X41.

Cf. A071062, A071070.

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.

A114835 Minimal set of palindrome prime-strings in base 10 in the sense of A071062.

Original entry on oeis.org

2, 3, 5, 7, 11, 919, 94049, 94649, 94849, 94949, 96469, 98689, 9809089, 9888889, 9889889, 9908099, 9980899, 9989899, 900808009, 906686609, 906989609, 908000809, 908444809, 908808809, 909848909, 960898069, 968999869, 988000889

Offset: 1

Walter Kehowski, Feb 19 2006

The one-digit primes are palindrome by default so the sequence starts off 2, 3, 5, 7 and so from now on all primes fitting one of the patterns *2*, *3*, *5*, *7* are excluded. The first palindrome prime that is not excluded is 11 so our list is now 2, 3, 5, 7, 11. Observe that from now on all the only palindrome primes allowed have first and last digit 9 and the only middle digits allowed are then 0, 1, 4, 6, 8, 9. But the first and only 3 digit palindrome to occur is 919, so from now on the only middle digits allowed are 0, 4, 6, 8, 9. The five digit palindrome primes that do not fit the patterns *2*, *3*, *5*, *7*, *1*1* and *9*1*9* are 94049, 94649, 94849, 94949, 96469, 98689. The rest of the sequence is determined in this recursive manner.

			a(6)=919 since it is the first 3-digit palindrome prime that does not fit the wildcard pattern established by the previous 5 elements, 2, 3, 5, 7, 11.

J.-P. Delahaye, "Pour la science", (French edition of Scientific American), Juin 2002, p. 99.
J. Shallit, Minimal primes, J. Recreational Mathematics, vol. 30.2, pp. 113-117, 1999-2000.

Walter A. Kehowski, Full list of terms
C. K. Caldwell, The Prime Glossary, minimal prime
J. Shallit, Minimal primes, J. Recreational Mathematics, vol. 30.2, pp. 113-117, 1999-2000.

Cf. A071062, A071070.

Maple
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Maple worksheet available upon request.
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A320726 Composite numbers such that all other numbers obtained from all permutations of all subsets of the digits are noncomposite.

Original entry on oeis.org

4, 6, 8, 9, 10, 20, 22, 30, 32, 33, 35, 50, 55, 70, 77, 111

Offset: 1

Daniel Lignon, Oct 19 2018

Sequence is finite since it is a subsequence of a finite sequence (A071070).

This is complete: there are only 16 terms in the sequence.

			371 is in this sequence because it's composite and none of the numbers 1, 3, 7, 13, 17, 31, 37, 137, 173, 317, 713 and 731 is composite.

Subsequence of A071070. Cf. A320725 (the same for prime numbers).

Select[Range[4, 10^3], Function[n, And[CompositeQ@ n, NoneTrue[DeleteCases[Flatten@ Map[If[Length@ # > 1, FromDigits /@ Permutations@ #, #] &, Rest@ Subsets@ IntegerDigits@ n], ?(# == n &)], CompositeQ]]]] (* _Michael De Vlieger, Nov 13 2018 *)

cp's OEIS Frontend

A110615 Minimal set of composite-strings in base 12 in the sense of A071070.

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A071062 Minimal set of prime-strings in base 10.

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A071071 Minimal "powers of 2" set in base 10: any power of 2 contains at least one term of this sequence in its decimal expansion.

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A071073 Minimal "multiples of 3" set in base 10.

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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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A071072 Minimal "multiples of 4" set in base 10.

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A110600 Minimal set of prime-strings in base 12 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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Maple

A114835 Minimal set of palindrome prime-strings in base 10 in the sense of A071062.

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Maple

A320726 Composite numbers such that all other numbers obtained from all permutations of all subsets of the digits are noncomposite.

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