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

%I A347819 #45 Apr 21 2023 02:07:47
%S A347819 11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,227,
%T A347819 251,257,277,281,349,409,449,499,521,557,577,587,727,757,787,821,827,
%U A347819 857,877,881,887,991,2087,2221,5051,5081,5501,5581,5801,5851,6469,6949,8501
%N A347819 Minimal elements for the base-10 representations of the primes greater than 10.
%C A347819 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).
%C A347819 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)
%C A347819 b=2: {11}
%C A347819 b=3: {12, 21, 111}
%C A347819 b=4: {11, 13, 23, 31, 221}
%C A347819 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}
%C A347819 b=6: {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}
%C A347819 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)
%C A347819 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}.
%C A347819 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
%H A347819 Jinyuan Wang, <a href="/A347819/b347819.txt">Table of n, a(n) for n = 1..77</a>
%H A347819 Curtis Bright, Raymond Devillers, and Jeffrey Shallit, <a href="https://doi.org/10.1080/10586458.2015.1064048">Minimal Elements for the Prime Numbers</a>, Experimental Mathematics 25 (3) (2016), pp. 321-331. DOI:10.1080/10586458.2015.1064048
%H A347819 Eric Chen, <a href="/A347819/a347819.pdf">Minimal elements for the base b representations of the primes which are > b</a>
%H A347819 Eric Chen, <a href="/A347819/a347819.txt">Data for minimal elements for the base b representations of the primes which are > b for 2 <= b <= 16</a>
%H A347819 Eric Chen, <a href="/A347819/a347819_1.txt">Proof for that the data for bases 2, 3, 4, 5, 6, 8, 10, 12 is complete</a>
%H A347819 Eric Chen, <a href="/A347819/a347819_5.txt">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</a>
%H A347819 Eric Chen, <a href="/A347819/a347819_4.txt">Condensed table for the status of the minimal element for the base b representations of the primes which are > b for 2 <= b <= 16</a>
%H A347819 Prime Glossary, <a href="https://primes.utm.edu/glossary/xpage/MinimalPrime.html">Minimal prime</a>
%H A347819 J. Shallit, <a href="https://cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf">Minimal primes</a>, J. Recreational Math., 30:2 (2000) pp. 113-117.
%e A347819 277 is in this sequence because none of 2, 7, 27, 77 is a prime > 10.
%e A347819 857 is in this sequence because none of 8, 5, 7, 85, 87, 57 is a prime > 10.
%e A347819 991 is in this sequence because none of 9, 1, 99, 91 is a prime > 10.
%e A347819 149 is not in this sequence because 19 is subsequence of 149 and 19 is a prime > 10.
%e A347819 389 is not in this sequence because 89 is subsequence of 389 and 89 is a prime > 10.
%e A347819 439 is not in this sequence because 43 is subsequence of 439 and 43 is a prime > 10.
%o A347819 (PARI) 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)
%o A347819 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
%o A347819 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.
%o A347819 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
%Y A347819 Cf. A071062 (primes > 10 are not required).
%Y A347819 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).
%K A347819 nonn,base,fini,full
%O A347819 1,1
%A A347819 _Eric Chen_, Sep 16 2021
%E A347819 Edited by _M. F. Hasler_, May 03 2022