A356737 Number of integer partitions of n into odd parts covering an interval of odd numbers.

1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 7, 8, 9, 10, 13, 13, 15, 17, 19, 21, 25, 26, 29, 33, 37, 40, 46, 49, 54, 61, 66, 72, 81, 87, 97, 106, 115, 125, 139, 150, 163, 179, 193, 210, 232, 248, 269, 293, 317, 343, 373, 401, 433, 470, 507, 545, 590, 633, 682, 737, 790

Offset: 0

Author

Gus Wiseman, Sep 03 2022

Keywords

• nonn

Examples

			The a(1) = 1 through a(9) = 6 partitions:
  1  11  3    31    5      33      7        53        9
         111  1111  311    3111    331      3311      333
                    11111  111111  31111    311111    531
                                   1111111  11111111  33111
                                                      3111111
                                                      111111111
		

Crossrefs

The strict case is A034178, for compositions A332032.

The initial case is A053251, ranked by A356232 and A356603.

The initial case for compositions is A356604.

The version for compositions is A356605, ranked by A060142 /\ A356841.

A000041 counts partitions, compositions A011782.

A066208 lists numbers with all odd prime indices, counted by A000009.

A073491 lists gapless numbers, initial A055932.

Cf. A001227, A066205, A107428, A107429, A333217, A356224, A356846.

Programs

Mathematica

nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
Table[Length[Select[IntegerPartitions[n],And@@OddQ/@#&&nogapQ[(#+1)/2]&]],{n,0,30}]