A071072 -id:A071072 - cp's OEIS frontend

A071062 Minimal set of prime-strings in base 10.

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

A071070 Minimal set of composite-strings in base 10.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 15, 20, 21, 22, 25, 27, 30, 32, 33, 35, 50, 51, 52, 55, 57, 70, 72, 75, 77, 111, 117, 171, 371, 711, 713, 731

Offset: 1

Benoit Cloitre, May 26 2002

Any composite number contains in its digits at least one of the term of this sequence and there is no smaller set.

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

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

Cf. A071062, A071071, A071072, A071073.

subs[digits_List] := Select[Subsets[digits], CompositeQ[FromDigits[#]]&] //. {a___List, b_List, c___List, d_List, e___List} /; MemberQ[Subsets[d], b] :> {a, b, c, e};
aa = {};
Do[aa = Union[aa, subs[IntegerDigits[n]]], {n, Select[Range[1000], CompositeQ]}];
A071070 = FromDigits /@ aa (* Jean-François Alcover, Dec 20 2017 *)

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.

A347819 Minimal elements for the base-10 representations of the primes greater than 10.

Original entry on oeis.org

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501

Offset: 1

Eric Chen, Sep 16 2021

Sequence is finite with 77 terms, the largest being 5*10^30 + 27 (which can be written 5(0_28)27, where 0_28 means the string of 28 0's). See text file for proof (this file also has proofs for bases 2, 3, 4, 5, 6, 8, 12).
Minimal elements for the base b representations of the primes > b for other bases b: (see the text file for 9 <= b <= 16) (all written in base b)
b=2: {11}
b=3: {12, 21, 111}
b=4: {11, 13, 23, 31, 221}
b=5: {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 10^95 + 13}
b=6: {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}
b=7: {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331} (conjectured, not proven)
b=8: {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, (10^220-1)/9*40 + 7}.
Equivalently: primes > 10 such that no proper substring (i.e., deleting any positive number of digits) is again a prime > 10. - M. F. Hasler, May 03 2022

			277 is in this sequence because none of 2, 7, 27, 77 is a prime > 10.
857 is in this sequence because none of 8, 5, 7, 85, 87, 57 is a prime > 10.
991 is in this sequence because none of 9, 1, 99, 91 is a prime > 10.
149 is not in this sequence because 19 is subsequence of 149 and 19 is a prime > 10.
389 is not in this sequence because 89 is subsequence of 389 and 89 is a prime > 10.
439 is not in this sequence because 43 is subsequence of 439 and 43 is a prime > 10.

Jinyuan Wang, Table of n, a(n) for n = 1..77
Curtis Bright, Raymond Devillers, and Jeffrey Shallit, Minimal Elements for the Prime Numbers, Experimental Mathematics 25 (3) (2016), pp. 321-331. DOI:10.1080/10586458.2015.1064048
Eric Chen, Minimal elements for the base b representations of the primes which are > b
Eric Chen, Data for minimal elements for the base b representations of the primes which are > b for 2 <= b <= 16
Eric Chen, Proof for that the data for bases 2, 3, 4, 5, 6, 8, 10, 12 is complete
Eric Chen, Known values or lower bounds of the largest minimal element for the base b representations of the primes which are > b for 2 <= b <= 36
Eric Chen, Condensed table for the status of the minimal element for the base b representations of the primes which are > b for 2 <= b <= 16
Prime Glossary, Minimal prime
J. Shallit, Minimal primes, J. Recreational Math., 30:2 (2000) pp. 113-117.

Cf. A071062 (primes > 10 are not required).

Minimal sets for other sets: A071070 (for composites), A071071 (powers of 2), A071072 (multiples of 4), A071073 (multiples of 3), A111055 (primes of the form 4*n+1), A111056 (primes of the form 4*n+3), A114835 (palindromic primes), A130448 (minimal set of squares).

a(n, k, b)=v=[]; for(r=1, length(digits(n, b)), if(r+length(digits(k, 2))-length(digits(n, b))>0 && digits(k, 2)[r+length(digits(k, 2))-length(digits(n, b))]==1, v=concat(v, digits(n, b)[r]))); fromdigits(v, b)
iss(n, b)=for(k=1, 2^length(digits(n, b))-2, if(ispseudoprime(a(n, k, b)) && a(n, k, b)>b, return(0))); 1
is(n, b=10)=isprime(n) && n>b && iss(n, b) \\ Test whether n is a minimal element for the base b representations of the primes > b. Default value b = 10 for this sequence.
select( {is_A347819(n,b=10)=for(L=2, #n=digits(n,b), forvec(d=vector(L, i, [1,#n]), n[d[1]]&& isprime(fromdigits(vecextract(n,d),b))&& return(L==#n), 2))}, [1..8888]) \\ Better select among primes([1,N]). - M. F. Hasler, May 03 2022

Edited by M. F. Hasler, May 03 2022

cp's OEIS Frontend

A071062 Minimal set of prime-strings in base 10.

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A071070 Minimal set of composite-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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A347819 Minimal elements for the base-10 representations of the primes greater than 10.

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A172982 Partial sums of minimal set of prime-strings in base 10 (A071062).

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