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A384084 Numbers whose prime signatures are self-conjugate.

%I A384084 #21 May 31 2025 21:53:38
%S A384084 1,2,3,5,7,11,12,13,17,18,19,20,23,28,29,31,36,37,41,43,44,45,47,50,
%T A384084 52,53,59,61,63,67,68,71,73,75,76,79,83,89,92,97,98,99,100,101,103,
%U A384084 107,109,113,116,117,120,124,127,131,137,139,147,148,149,151,153
%N A384084 Numbers whose prime signatures are self-conjugate.
%C A384084 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), and 2 + 1 is a self-conjugate partition of 3. In contrast, while bigomega(42) = 3, 3 is partitioned as 1 + 1 + 1, because 42 = (2^1)(3^1)(7^1), and 1 + 1 + 1 is not a self-conjugate partition of 3.
%C A384084 This sequence is very similar to, but ultimately different from, A212166. The first difference is a(342) = 1083, whereas A212166(342) = 1080.
%C A384084 This sequence is a subsequence of A212166.
%C A384084 It includes 1 (empty partition) and all primes (A000040: partition 1), as well as numbers of the form (p^2)q, where p and q are distinct primes (A054753: partition 2 + 1).
%C A384084 k is a term in this sequence if and only if A046523(k) is a term in A181825.
%H A384084 Hal M. Switkay, <a href="/A384084/b384084.txt">Table of n, a(n) for n = 1..342</a>
%H A384084 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Self-ConjugatePartition.html">Self-Conjugate Partition</a>.
%e A384084 120 is a term; its prime factorization (2^3)(3^2)(5^1) is self-conjugate.
%e A384084 24 is not a term; its prime factorization (2^3)(3^1) is not self-conjugate.
%t A384084 selfQ[p_] := ResourceFunction["ConjugatePartition"][p] == p; q[n_] := selfQ[Sort[FactorInteger[n][[;;, 2]], Greater]]; Select[Range[200], q] (* _Amiram Eldar_, May 26 2025 *)
%Y A384084 Cf. A001222, A046523, A054753, A181825, A212166.
%K A384084 nonn
%O A384084 1,2
%A A384084 _Hal M. Switkay_, May 18 2025