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Row lengths of A276427: largest k such that a partition of n has k-1 distinct parts i of multiplicity i.
2

%I #10 Oct 28 2019 12:15:28

%S 1,2,1,2,2,3,2,2,3,3,3,3,3,3,4,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,4,4,4,

%T 4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,5,5,5,5,5,6,6,6,6,6,6,6,

%U 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,6,6,6,6,6,6,7,7,7,7,7,7,7,7

%N Row lengths of A276427: largest k such that a partition of n has k-1 distinct parts i of multiplicity i.

%C Columns of A276427 are numbered starting with 0, so the row length is one more than the index of the last column.

%e For n = 0, the empty partition [] has 0 parts i with multiplicity i, so a(0) = 1.

%e For n = 1, the partition [1] has one part i with multiplicity i, whence a(1) = 2.

%e For n = 2, both partitions [1,1] and [2] have 0 parts i with multiplicity i, so a(2) = 1.

%e For n = 3, the partition [1,2] has one part i with multiplicity i, hence a(3) = 2.

%e For n = 4, the partitions [1,3] and [2,2] have one part i with multiplicity i, so a(4) = 2.

%e For n = 5, the partition [1,2,2] has 2 parts i with multiplicity i, hence a(5) = 3.

%e The smallest partition with k-1 = 3 parts i with multiplicity i is [1,2,2,3,3,3], for n = 14, whence a(14) = 4.

%o (PARI) a(n)=#A276427_row(n)

%Y Cf. A276427, A276428, A276429, A000041.

%K nonn

%O 0,2

%A _M. F. Hasler_, Oct 27 2019

%E More terms from _Alois P. Heinz_, Oct 28 2019