A373290 a(1) = 1. Thereafter, for n prime a(n) is the smallest composite number not already a term which is not divisible by n, and for n composite a(n) is the smallest prime not already a term, which does not divide n.

1, 9, 4, 3, 6, 5, 8, 7, 2, 11, 10, 13, 12, 17, 19, 23, 14, 29, 15, 31, 37, 41, 16, 43, 47, 53, 59, 61, 18, 67, 20, 71, 73, 79, 83, 89, 21, 97, 101, 103, 22, 107, 24, 109, 113, 127, 25, 131, 137, 139, 149, 151, 26, 157, 163, 167, 173, 179, 27, 181, 28, 191, 193

Offset

1,2

Comments

{a(1),a(2),...,a(9)} is a self-inverse permutation of the first 9 terms of A026239, and for n >= 10 a(n) = A026239(n). Since A026239 is a self-inverse permutation of the natural numbers, so is this sequence (but primes < 11 are not in order).

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^4, showing primes in red and nonprimes in blue.

Example

For n = 2, (prime), a(2) = 9, the smallest composite number not divisible by 2.
For n = 6, (composite), a(6) = 5, the smallest novel prime which does not divide 6.

Mathematica

nn = 120; c[_] := False; a[1] = 1; c[1] = True; u = 2; v = 4;
Do[If[PrimeQ[n],
  k = v; While[Or[c[k], PrimeQ[k], Divisible[k, n]], k++],
  k = u; While[Or[c[k], CompositeQ[k], Divisible[n, k]], k++]];
 Set[{a[n], c[k]}, {k, True}];
 If[k == u, While[Or[c[u], CompositeQ[u]], u++]];
 If[k == v, While[Or[c[v], PrimeQ[v]], v++]], {n, 2, nn}];
Array[a, nn] (* Michael De Vlieger, May 31 2024 *)

Crossrefs

Cf. A000027, A000040, A002808, A026239.

Sequence in context: A010540 A187466 A082695 * A236257 A019909 A324002

Adjacent sequences: A373287 A373288 A373289 * A373291 A373292 A373293

Keyword

nonn

Author

David James Sycamore, May 30 2024

Extensions

More terms from Michael De Vlieger, May 31 2024

Status

approved