OFFSET
1,1
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
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
Prime Glossary, Minimal prime
J. Shallit, Minimal primes, J. Recreational Math., 30:2 (2000) pp. 113-117.
EXAMPLE
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.
PROG
(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
CROSSREFS
Cf. A071062 (primes > 10 are not required).
KEYWORD
nonn,base,fini,full
AUTHOR
Eric Chen, Sep 16 2021
EXTENSIONS
Edited by M. F. Hasler, May 03 2022
STATUS
approved