A088629 -id:A088629 - cp's OEIS frontend

A088630 Members of A088629 such that all the eligible odd numbers less than n have already occurred.

1, 3, 7, 9, 11, 17, 23, 31, 37, 39, 43, 51, 53, 67, 71, 73, 77, 89, 97, 101, 107, 137, 143, 159, 177, 183, 199, 209, 211, 217, 221, 231, 233, 237, 243, 247, 249, 277, 281, 283, 299, 303, 311, 313, 319, 323, 331, 363, 377, 381, 401, 417, 423, 449, 453, 481, 489

Offset: 0

Amarnath Murthy, Oct 21 2003

"Eligible" means not divisible by 5.

			9 is a member but 13, 19 etc. are not members of this sequence.
13 is not a member because 11 doesn't occur before 13 in A088629.

Cf. A088629.

More terms from David Wasserman, Aug 16 2005

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.

Original entry on oeis.org

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

Rémy Sigrist, Jan 20 2023

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.

			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}

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

Cf. A083569, A088629.

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 *)

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A088630 Members of A088629 such that all the eligible odd numbers less than n have already occurred.

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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.

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PARI