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%I #11 Oct 24 2023 15:42:49
%S 1,2,1,2,1,1,2,1,6,1,2,1,6,2,1,2,1,6,2,8,1,2,1,6,2,8,1,1,2,1,6,2,8,1,
%T 6,1,2,1,6,2,8,1,6,22,1,2,1,6,2,8,1,6,22,2,1,2,1,6,2,8,1,6,22,2,8,1,2,
%U 1,6,2,8,1,6,22,2,8,30,1,2,1,6,2,8,1,6,22
%N a(n) = number of partitions of n that have the maximum number of neighbors; see Comments.
%C Partitions p and q of n are neighbors if d(p,q) = 2, where d is the distance function in A366156.
%e Refer to the Example in A366429 to see that a(5) = 1.
%t c[n_] := PartitionsP[n];
%t q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
%t r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
%t d[u_, v_] := d[u, v] = Total[Abs[u - v]];
%t s[n_, k_] := s[n, k] = Select[Range[c[n]], d[r[n, k], r[n, #]] == 2 &]
%t t[n_] := t[n] = Table[s[n, k], {k, 1, c[n]}]
%t a[n_] := Max[Map[Length, t[n]]]
%t b[n_] := b[n] = Select[t[n], Length[#] == a[n] &]
%t e[n_] := Length[b[n]]
%t Table[e[n], {n, 1, 24}]
%Y Cf. A000041, A366156, A366429.
%K nonn
%O 1,2
%A _Clark Kimberling_, Oct 12 2023
%E More terms from _Pontus von Brömssen_, Oct 24 2023
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