A228323 a(1)=1; thereafter a(n) is the smallest number m not yet in the sequence such that at least one of the concatenations a(n-1)||m or m||a(n-1) is prime.

1, 3, 2, 9, 5, 21, 4, 7, 6, 13, 10, 19, 16, 27, 8, 11, 15, 23, 12, 17, 20, 29, 14, 33, 26, 47, 18, 31, 25, 39, 22, 37, 24, 41, 30, 49, 34, 57, 28, 43, 36, 59, 32, 51, 38, 53, 42, 61, 45, 67, 58, 69, 55, 63, 44, 81, 35, 71, 48, 77, 50, 87, 62, 99, 40, 73, 46

Offset

1,2

Comments

Does every number appear in the sequence?

If a(n) is coprime to 10, then a(n+1) exists by Dirichlet's theorem. - Eric M. Schmidt, Aug 20 2013 [In more detail: let a(n) have d digits, and consider the arithmetic progression k*10^d + a(n), and apply Dirichlet's theorem. This gives a number k such that the concatenation k||a(n) is prime. N. J. A. Sloane, Nov 08 2020]

The argument in A068695 shows that a(n) always exists. - N. J. A. Sloane, Nov 11 2020

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Eric Angelini, Primes by concatenation, Posting to the Sequence Fans Mailing List, Aug 14 2013.

Michael De Vlieger, Labeled log-log scatterplot of a(n) n = 1..2^14, showing m coprime to 10 in red, otherwise dark blue.

Index entries for primes involving decimal expansion of n

Mathematica

f[s_] := Block[{k = 2, idj = IntegerDigits@ s[[-1]]}, While[idk = IntegerDigits@ k; MemberQ[s, k] || ( !PrimeQ@ FromDigits@ Join[idj, idk] && !PrimeQ@ FromDigits@ Join[idk, idj]), k++]; Append[s, k]]; Nest[f, {1}, 66] (* Robert G. Wilson v, Aug 20 2013 *)

Prog

(Python)
from sympy import isprime
from itertools import islice
def c(s, t): return isprime(int(s+t)) or isprime(int(t+s))
def agen():
    aset, k, mink = set(), 1, 2
    while True:
        an = k; aset.add(an); yield an; s, k = str(an), mink
        while k in aset or not c(s, str(k)): k += 1
        while mink in aset: mink += 1
print(list(islice(agen(), 56))) # Michael S. Branicky, Oct 17 2022

Crossrefs

See A228324 for the primes that arise.

Cf. A069695, A228325.

Sequence in context: A033313 A231442 A319107 * A350831 A140590 A329211

Adjacent sequences: A228320 A228321 A228322 * A228324 A228325 A228326

Keyword

nonn,base

Author

N. J. A. Sloane, Aug 20 2013

Extensions

More terms from Alois P. Heinz, Aug 20 2013

Status

approved