A380944 a(n) = b(n,A000120(n)) for n >= 0 where b(n,k) is defined in Comments.
1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 9, 3, 4, 1, 5, 4, 7, 3, 13, 5, 7, 2, 23, 9, 12, 3, 16, 4, 5, 1, 6, 5, 9, 4, 17, 7, 10, 3, 31, 13, 18, 5, 25, 7, 9, 2, 53, 23, 32, 9, 44, 12, 15, 3, 64, 16, 20, 4, 25, 5, 6, 1, 7, 6, 11, 5, 21, 9, 13, 4, 39, 17, 24, 7, 34, 10
Offset: 0
Programs
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PARI
b(n,k) = if(k==0, 1, if(n==0, 0, if(n%2, b((n-1)/2,k) + b((n-1)/2,k-1), (hammingweight(n/2)-k+1)*b(n+1,k)))) a(n) = b(n, hammingweight(n))
Formula
Conjecture: a(2^m*(2^n-1)) = (n+1)^m - n*n!*c(m,n+1) for n >= 0, m >= 0 where c(n,k) = Sum_{i=0..n-k} Stirling2(k+i,k) for n >= 0, k >= 0.
Comments