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.
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.
These are the positions of 1's in A257991.
The even prime indices are counted by A257992.
The conjugate partitions are ranked by A345958.
A122111 is a representation of partition conjugation.
A349157 ranks partitions with as many even parts as odd conjugate parts.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 12 2021
STATUS
approved