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

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

Author

Eric Chen, Sep 16 2021

Keywords

• nonn
• base
• fini
• full

Comments

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

Examples

			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.
		

Links

• 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.

Crossrefs

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).

Programs

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)
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

Extensions

Edited by M. F. Hasler, May 03 2022