OFFSET
1,1
COMMENTS
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
EXAMPLE
The sequence of partitions together with their Heinz numbers begins:
2: (1) 59: (17) 120: (3,2,1,1,1)
5: (3) 66: (5,2,1) 124: (11,1,1)
8: (1,1,1) 67: (19) 125: (3,3,3)
11: (5) 68: (7,1,1) 127: (31)
17: (7) 73: (21) 128: (1,1,1,1,1,1,1)
20: (3,1,1) 75: (3,3,2) 137: (33)
23: (9) 80: (3,1,1,1,1) 138: (9,2,1)
30: (3,2,1) 83: (23) 149: (35)
31: (11) 92: (9,1,1) 153: (7,2,2)
32: (1,1,1,1,1) 97: (25) 154: (5,4,1)
41: (13) 99: (5,2,2) 157: (37)
44: (5,1,1) 102: (7,2,1) 164: (13,1,1)
45: (3,2,2) 103: (27) 165: (5,3,2)
47: (15) 109: (29) 167: (39)
50: (3,3,1) 110: (5,3,1) 170: (7,3,1)
MATHEMATICA
Select[Range[100], OddQ[PrimeOmega[#]*PrimePi[FactorInteger[#][[-1, 1]]]]&]
CROSSREFS
Note: Heinz numbers are given in parentheses below.
The case of odd length only is A026424.
The case of odd maximum only is A244991.
Positions of odd terms in A326846.
These partitions are counted by A340385.
The version for factorizations is A340607.
A027193 counts partitions of odd length, or of odd maximum.
A061395 gives maximum prime index.
A106529 lists numbers with Omega equal to maximum prime index.
A339890 counts factorizations of odd length.
A340102 counts odd-length factorizations into odd factors.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 25 2021
STATUS
approved