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

Author

David James Sycamore, May 30 2024

Keywords

• nonn

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).

Examples

			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.
		

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.

Crossrefs

Cf. A000027, A000040, A002808, A026239.

Programs

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 *)

Extensions

More terms from Michael De Vlieger, May 31 2024