This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384564 #8 Jun 09 2025 23:59:50
%S A384564 1,2,3,6,5,4,7,30,10,9,11,12,13,25,49,210,17,18,19,20,121,169,23,60,
%T A384564 14,289,42,28,29,8,31,2310,361,529,841,36,37,961,1369,84,41,27,43,44,
%U A384564 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
%N A384564 a(n) = the least natural number not already in the sequence whose prime signature is conjugate to that of n.
%C A384564 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).
%e A384564 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.
%t A384564 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 *)
%Y A384564 Cf. A000040, A000079, A001248, A002110, A006881, A384084.
%K A384564 nonn
%O A384564 1,2
%A A384564 _Hal M. Switkay_, Jun 03 2025