A384084 Numbers whose prime signatures are self-conjugate.

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 36, 37, 41, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 109, 113, 116, 117, 120, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153

Offset: 1

Author

Hal M. Switkay, May 18 2025

Keywords

• nonn

Comments

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.

This sequence is very similar to, but ultimately different from, A212166. The first difference is a(342) = 1083, whereas A212166(342) = 1080.

This sequence is a subsequence of A212166.

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).

k is a term in this sequence if and only if A046523(k) is a term in A181825.

Examples

			120 is a term; its prime factorization (2^3)(3^2)(5^1) is self-conjugate.
24 is not a term; its prime factorization (2^3)(3^1) is not self-conjugate.
		

Links

• Hal M. Switkay, Table of n, a(n) for n = 1..342
• Eric Weisstein's World of Mathematics, Self-Conjugate Partition.

Crossrefs

Cf. A001222, A046523, A054753, A181825, A212166.

Programs

Mathematica

selfQ[p_] := ResourceFunction["ConjugatePartition"][p] == p; q[n_] := selfQ[Sort[FactorInteger[n][[;;, 2]], Greater]]; Select[Range[200], q] (* Amiram Eldar, May 26 2025 *)