A340611 Number of integer partitions of n of length 2^k where k is the greatest part.

1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 6, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 29, 32, 34, 36, 38, 41, 42, 45, 47, 50, 52, 56, 58, 63, 66, 71, 75, 83, 88, 98, 106, 118, 128, 143, 155, 173, 188, 208, 226, 250, 270, 297, 321, 350

Offset: 0

Author

Gus Wiseman, Jan 28 2021

Keywords

• nonn

Comments

Also the number of integer partitions of n with maximum 2^k where k is the length.

Examples

			The partitions for n = 12, 14, 16, 22, 24:
  32211111  32222111  32222221  33333322          33333333
  33111111  33221111  33222211  33333331          4222221111111111
            33311111  33322111  4222111111111111  4322211111111111
                      33331111  4321111111111111  4332111111111111
                                4411111111111111  4422111111111111
                                                  4431111111111111
The conjugate partitions:
  (8,2,2)  (8,3,3)  (8,4,4)  (8,7,7)     (8,8,8)
  (8,3,1)  (8,4,2)  (8,5,3)  (8,8,6)     (16,3,3,2)
           (8,5,1)  (8,6,2)  (16,2,2,2)  (16,4,2,2)
                    (8,7,1)  (16,3,2,1)  (16,4,3,1)
                             (16,4,1,1)  (16,5,2,1)
                                         (16,6,1,1)
		

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.

A018818 counts partitions of n into divisors of n (A326841).

A047993 counts balanced partitions (A106529).

A067538 counts partitions of n whose length/max divides n (A316413/A326836).

A072233 counts partitions by sum and length.

A168659 = partitions whose greatest part divides their length (A340609).

A168659 = partitions whose length divides their greatest part (A340610).

A326843 = partitions of n whose length and maximum both divide n (A326837).

A340597 lists numbers with an alt-balanced factorization.

A340653 counts balanced factorizations.

A340689 have a factorization of length 2^max.

A340690 have a factorization of maximum 2^length.

Cf. A003114, A006141, A064174, A098124, A117409, A200750, A340385, A340387, A340599, A340601.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Length[#]==2^Max@@#&]],{n,0,30}]