A359973 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, the concatenation of the decimal digits of n and a(n) or of a(n) and n yields a prime number.

1, 3, 2, 7, 9, 13, 4, 11, 5, 19, 8, 17, 6, 23, 31, 21, 12, 47, 10, 27, 16, 37, 14, 41, 39, 33, 20, 43, 32, 49, 15, 29, 26, 57, 59, 71, 22, 51, 25, 73, 24, 53, 28, 63, 61, 79, 18, 77, 30, 81, 38, 97, 42, 83, 69, 89, 34, 67, 35, 91, 45, 87, 44, 109, 99, 103, 58

Offset

1,2

Comments

Leading zeros are ignored.

This sequence is a self-inverse permutation of the positive integers (for any positive number v, there are infinitely many prime numbers starting with 10*v+1, so infinitely many prime numbers that are the concatenation of v and some other positive integer).

There is only one fixed point: a(1) = 1.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..16384

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing a(n) coprime to 10 in red and other numbers in dark blue.

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^10, showing primes in red, composite prime powers in gold, squarefree composites in dark green, and other numbers in blue, showing powerful numbers that are not prime powers in light blue.

Mathematics StackExchange, Proof that there are infinitely many prime numbers starting with a given digit string

Rémy Sigrist, PARI program

Index entries for sequences that are permutations of the natural numbers

Example

The first terms, alongside the corresponding prime numbers, are:
  n   a(n)  Corresponding prime numbers
  --  ----  ---------------------------
   1     1  {11}
   2     3  {23}
   3     2  {23}
   4     7  {47}
   5     9  {59}
   6    13  {613}
   7     4  {47}
   8    11  {811}
   9     5  {59}
  10    19  {1019}
  11     8  {811}
  12    17  {1217}

Mathematica

nn = 120; c[_] := False; a[1] = 1; c[1] = True; u = 2; Q[n_] := AnyTrue[{FromDigits[Join[d, #]], FromDigits[Join[#, d]]} & @@ {IntegerDigits[n], d}, PrimeQ]; Do[Set[{k, d}, {u, IntegerDigits[n]}]; While[Nand[! c[k], Q[k]], k++]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Jan 21 2023 *)

Prog

(PARI) See Links section.

Crossrefs

Cf. A083569, A088629.

Sequence in context: A118966 A351904 A351903 * A348535 A018891 A034423

Adjacent sequences: A359970 A359971 A359972 * A359974 A359975 A359976

Keyword

nonn,base

Author

Rémy Sigrist, Jan 20 2023

Status

approved