A375015 Lexicographically earliest sequence of distinct positive terms such that either the concatenation [a(n); n; a(n+1)] or the concatenation [a(n+1); n; a(n)] is a prime number.

1, 2, 3, 4, 9, 6, 19, 8, 7, 10, 21, 11, 17, 13, 5, 39, 16, 29, 14, 31, 20, 27, 24, 57, 23, 32, 33, 26, 53, 12, 49, 35, 37, 15, 43, 22, 67, 30, 59, 18, 73, 28, 61, 41, 51, 70, 47, 25, 69, 34, 89, 42, 79, 38, 71, 52, 77, 48, 93, 45, 91, 44, 63, 58, 99, 36, 83, 50, 87, 55

Offset

1,2

Comments

The sequence is a rearrangement of the positive integers.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Eric Angelini, Rightmost digit, leftmost digit, variants, Personal blog of the author.

Example

The first 11 terms are 1,2,3,4,9,6,19,8,7,10,21. Successively inserting n between a(n) and a(n+1) produce:
112, 223, 334, 449, 956, 6619, 1978, 887, 7910, 101021.
If such a concatenation is composite, the concatenation [a(n+1);n;a(n)] is prime by construction.
112, for instance, is not prime but 211 is. The same for 334 (composite) and 433 (prime), or 956 (composite) and 659 (prime). 7910 is not prime, but (0)197 is prime. If both concatenations are prime, we keep the smallest term.

Prog

(Python)
from sympy import isprime
from itertools import count, islice
def c(s, t, u): return isprime(int(s+t+u)) or isprime(int(u+t+s))
def agen(): # generator of terms
    an, aset, m = 1, set(), 2
    for n in count(1):
        yield an
        aset.add(an)
        s, t = str(an), str(n)
        an = next(k for k in count(m) if k not in aset and c(s, t, str(k)))
        while m in aset: aset.discard(m); m += 1
print(list(islice(agen(), 70)))

Crossrefs

Cf. A000040.

Sequence in context: A060866 A064478 A111798 * A249543 A307404 A307405

Adjacent sequences: A375012 A375013 A375014 * A375016 A375017 A375018

Keyword

base,nonn

Author

Eric Angelini and Michael S. Branicky, Aug 08 2024

Status

approved