%I #7 Feb 16 2021 09:35:37

%S 6,14,15,24,26,33,35,38,51,54,56,58,60,65,69,74,77,86,93,95,96,104,

%T 106,119,122,123,126,132,135,140,141,142,143,145,150,152,158,161,177,

%U 178,185,201,202,204,209,214,215,216,217,219,221,224,226,232,234,240

%N Heinz numbers of integer partitions of type OO.

%C 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.

%e The sequence of partitions together with their Heinz numbers begins:

%e 6: (2,1) 74: (12,1) 141: (15,2)

%e 14: (4,1) 77: (5,4) 142: (20,1)

%e 15: (3,2) 86: (14,1) 143: (6,5)

%e 24: (2,1,1,1) 93: (11,2) 145: (10,3)

%e 26: (6,1) 95: (8,3) 150: (3,3,2,1)

%e 33: (5,2) 96: (2,1,1,1,1,1) 152: (8,1,1,1)

%e 35: (4,3) 104: (6,1,1,1) 158: (22,1)

%e 38: (8,1) 106: (16,1) 161: (9,4)

%e 51: (7,2) 119: (7,4) 177: (17,2)

%e 54: (2,2,2,1) 122: (18,1) 178: (24,1)

%e 56: (4,1,1,1) 123: (13,2) 185: (12,3)

%e 58: (10,1) 126: (4,2,2,1) 201: (19,2)

%e 60: (3,2,1,1) 132: (5,2,1,1) 202: (26,1)

%e 65: (6,3) 135: (3,2,2,2) 204: (7,2,1,1)

%e 69: (9,2) 140: (4,3,1,1) 209: (8,5)

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],OddQ[Count[primeMS[#],_?EvenQ]]&&OddQ[Count[primeMS[#],_?OddQ]]&]

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

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

%Y The type EE version is A236913 (A340784).

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

%Y A000009 counts partitions into odd parts (A066208).

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

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

%Y A058695 counts partitions of odd numbers (A300063).

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

%Y A340101 counts factorizations into odd factors.

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

%Y A340601 counts partitions of even rank (A340602).

%Y A340692 counts partitions of odd rank (A340603).

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Feb 15 2021