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A349158
Heinz numbers of integer partitions with exactly one odd part.
3
2, 5, 6, 11, 14, 15, 17, 18, 23, 26, 31, 33, 35, 38, 41, 42, 45, 47, 51, 54, 58, 59, 65, 67, 69, 73, 74, 77, 78, 83, 86, 93, 95, 97, 98, 99, 103, 105, 106, 109, 114, 119, 122, 123, 126, 127, 135, 137, 141, 142, 143, 145, 149, 153, 157, 158, 161, 162, 167, 174
OFFSET
1,1
COMMENTS
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with exactly one odd prime index. These are also partitions whose conjugate partition has alternating sum equal to 1.
Numbers that are product of a term of A031368 and a term of A066207. - Antti Karttunen, Nov 13 2021
EXAMPLE
The terms and corresponding partitions begin:
2: (1) 42: (4,2,1) 86: (14,1)
5: (3) 45: (3,2,2) 93: (11,2)
6: (2,1) 47: (15) 95: (8,3)
11: (5) 51: (7,2) 97: (25)
14: (4,1) 54: (2,2,2,1) 98: (4,4,1)
15: (3,2) 58: (10,1) 99: (5,2,2)
17: (7) 59: (17) 103: (27)
18: (2,2,1) 65: (6,3) 105: (4,3,2)
23: (9) 67: (19) 106: (16,1)
26: (6,1) 69: (9,2) 109: (29)
31: (11) 73: (21) 114: (8,2,1)
33: (5,2) 74: (12,1) 119: (7,4)
35: (4,3) 77: (5,4) 122: (18,1)
38: (8,1) 78: (6,2,1) 123: (13,2)
41: (13) 83: (23) 126: (4,2,2,1)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Count[primeMS[#], _?OddQ]==1&]
CROSSREFS
These partitions are counted by A000070 up to 0's.
Allowing no odd parts gives A066207, counted by A000041 up to 0's.
Requiring all odd parts gives A066208, counted by A000009.
These are the positions of 1's in A257991.
The even prime indices are counted by A257992.
The conjugate partitions are ranked by A345958.
Allowing at most one odd part gives A349150, counted by A100824.
A047993 ranks balanced partitions, counted by A106529.
A056239 adds up prime indices, row sums of A112798.
A122111 is a representation of partition conjugation.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A325698 ranks partitions with as many even as odd parts, counted by A045931.
A340604 ranks partitions of odd positive rank, counted by A101707.
A340932 ranks partitions whose least part is odd, counted by A026804.
A349157 ranks partitions with as many even parts as odd conjugate parts.
Sequence in context: A045845 A002133 A092306 * A319242 A323398 A233865
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 12 2021
STATUS
approved