A340611 - OEIS

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

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OFFSET

0,13

COMMENTS

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

LINKS

Table of n, a(n) for n=0..68.

EXAMPLE

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)

MATHEMATICA

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

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.

Sequence in context: A122258 A332220 A263089 * A068509 A070319 A057142

Adjacent sequences: A340608 A340609 A340610 * A340612 A340613 A340614

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 28 2021

STATUS

approved