A384564 a(n) = the least natural number not already in the sequence whose prime signature is conjugate to that of n.

1, 2, 3, 6, 5, 4, 7, 30, 10, 9, 11, 12, 13, 25, 49, 210, 17, 18, 19, 20, 121, 169, 23, 60, 14, 289, 42, 28, 29, 8, 31, 2310, 361, 529, 841, 36, 37, 961, 1369, 84, 41, 27, 43, 44, 45, 1681, 47, 420, 15, 50, 1849, 52, 53, 90, 2209, 126, 2809, 3481, 59, 24, 61, 3721, 63, 30030, 4489, 125, 67, 68, 5041, 343, 71, 180

Offset: 1

Author

Hal M. Switkay, Jun 03 2025

Keywords

• nonn

Comments

This sequence is an involutory (period 2) permutation of the natural numbers. Its graph is symmetric in the line a(n) = n. It fixes all numbers with self-conjugate prime signatures (A384084), including 1 and the primes (A000040). It exchanges prime squares (A001248) and products of two distinct primes (A006881). It exchanges powers of 2 (A000079) with primorials (A002110).

Examples

			The implied partition corresponding to k is the partition of bigomega(k) (A001222) formed by the prime exponents. For example, bigomega(18) = 3, which is partitioned as 2 + 1, because 18 = (3^2)*(2^1). Because this is a self-conjugate partition, a(18) = 18. Similarly, bigomega(42) = 3, which is partitioned as 1 + 1 + 1, because 42 = (2^1)*(3^1)*(7^1). The conjugate partition is 3, so a(42) is the cube of a prime.
		

Crossrefs

Cf. A000040, A000079, A001248, A002110, A006881, A384084.

Programs

Mathematica

sig[n_] := Sort[FactorInteger[n][[;; , 2]], Greater]; conjp[p_] := ResourceFunction["ConjugatePartition"][p]; a[1] = 1; a[n_] := a[n] = Module[{p = conjp[sig[n]], k = 2}, While[!FreeQ[Array[a, n-1], k] || sig[k] != p, k++]; k]; Array[a, 100] (* Amiram Eldar, Jun 04 2025 *)