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
Examples
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}
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
Programs
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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 *)
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PARI
See Links section.
Comments