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Numbers with recursively self-conjugate partitions are given in A190899. See that sequence or the Keith reference for more details.
It is proved in the reference that this list is exhaustive.
Table of n, a(n) for n=1..45.
William J. Keith, Recursively Self-Conjugate Partitions, INTEGERS 11A, (2011) Article 12 (11 pages).
Sequence in context: A192111 A032402 A280848 * A216572 A020674 A243171
Adjacent sequences: A190897 A190898 A190899 * A190901 A190902 A190903
John W. Layman, May 23 2011