A356603 -id:A356603 - cp's OEIS frontend

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

Gus Wiseman, Sep 03 2022

nonn

			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

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.

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

A356956 Numbers k such that the k-th composition in standard order is a gapless interval (in increasing order).

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 16, 20, 32, 52, 64, 72, 128, 256, 272, 328, 512, 840, 1024, 1056, 2048, 2320, 4096, 4160, 8192, 10512, 16384, 16512, 17440, 26896, 32768, 65536, 65792, 131072, 135232, 148512, 262144, 262656, 524288, 672800, 1048576, 1049600, 1065088, 1721376

Offset: 1

Gus Wiseman, Sep 24 2022

nonn

An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms and corresponding intervals begin:
        0: ()
        1: (1)
        2: (2)
        4: (3)
        6: (1,2)
        8: (4)
       16: (5)
       20: (2,3)
       32: (6)
       52: (1,2,3)
       64: (7)
       72: (3,4)
      128: (8)
      256: (9)
      272: (4,5)
      328: (2,3,4)
      512: (10)
      840: (1,2,3,4)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
These compositions are counted by A001227.
An unordered version is A073485, non-strict A073491 (complement A073492).
The initial version is A164894, non-strict A356843 (unordered A356845).
The non-strict version is A356841, initial A333217, counted by A107428.
A066311 lists gapless numbers.
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
A356844 ranks compositions with at least one 1.
Cf. A053251, A055932, A073493, A132747, A137921, A286470, A356224, A356842.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
chQ[y_]:=Length[y]<=1||Union[Differences[y]]=={1};
Select[Range[0,1000],chQ[stc[#]]&]

cp's OEIS Frontend

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

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