A341448 Heinz numbers of integer partitions of type OO.

6, 14, 15, 24, 26, 33, 35, 38, 51, 54, 56, 58, 60, 65, 69, 74, 77, 86, 93, 95, 96, 104, 106, 119, 122, 123, 126, 132, 135, 140, 141, 142, 143, 145, 150, 152, 158, 161, 177, 178, 185, 201, 202, 204, 209, 214, 215, 216, 217, 219, 221, 224, 226, 232, 234, 240

Offset

1,1

Comments

These partitions are defined to have an odd number of odd parts and an odd number of even parts. They also have even length and odd sum.

Links

Table of n, a(n) for n=1..56.

Example

The sequence of partitions together with their Heinz numbers begins:
      6: (2,1)         74: (12,1)           141: (15,2)
     14: (4,1)         77: (5,4)            142: (20,1)
     15: (3,2)         86: (14,1)           143: (6,5)
     24: (2,1,1,1)     93: (11,2)           145: (10,3)
     26: (6,1)         95: (8,3)            150: (3,3,2,1)
     33: (5,2)         96: (2,1,1,1,1,1)    152: (8,1,1,1)
     35: (4,3)        104: (6,1,1,1)        158: (22,1)
     38: (8,1)        106: (16,1)           161: (9,4)
     51: (7,2)        119: (7,4)            177: (17,2)
     54: (2,2,2,1)    122: (18,1)           178: (24,1)
     56: (4,1,1,1)    123: (13,2)           185: (12,3)
     58: (10,1)       126: (4,2,2,1)        201: (19,2)
     60: (3,2,1,1)    132: (5,2,1,1)        202: (26,1)
     65: (6,3)        135: (3,2,2,2)        204: (7,2,1,1)
     69: (9,2)        140: (4,3,1,1)        209: (8,5)

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], OddQ[Count[primeMS[#], _?EvenQ]]&&OddQ[Count[primeMS[#], _?OddQ]]&]

Crossrefs

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

The case of odd parts, length, and sum is counted by A078408 (A300272).

The type EE version is A236913 (A340784).

These partitions (for odd n) are counted by A236914.

A000009 counts partitions into odd parts (A066208).

A026804 counts partitions whose least part is odd (A340932).

A027193 counts partitions of odd length/maximum (A026424/A244991).

A058695 counts partitions of odd numbers (A300063).

A160786 counts odd-length partitions of odd numbers (A340931).

A340101 counts factorizations into odd factors.

A340385 counts partitions of odd length and maximum (A340386).

A340601 counts partitions of even rank (A340602).

A340692 counts partitions of odd rank (A340603).

Cf. A000700, A024429, A027187, A106529, A117409, A174725, A257541, A325134, A339890, A340102, A340604.

Sequence in context: A192321 A190272 A107982 * A081535 A355941 A359912

Adjacent sequences: A341445 A341446 A341447 * A341449 A341450 A341451

Keyword

nonn

Author

Gus Wiseman, Feb 15 2021

Status

approved