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
Keywords
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.
Programs
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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 *)
Comments