A340386 Heinz numbers of integer partitions with an odd number of parts, the greatest of which is odd.

2, 5, 8, 11, 17, 20, 23, 30, 31, 32, 41, 44, 45, 47, 50, 59, 66, 67, 68, 73, 75, 80, 83, 92, 97, 99, 102, 103, 109, 110, 120, 124, 125, 127, 128, 137, 138, 149, 153, 154, 157, 164, 165, 167, 170, 176, 179, 180, 186, 188, 191, 197, 200, 207, 211, 227, 230

Offset: 1

Author

Gus Wiseman, Jan 25 2021

Keywords

• nonn

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.

Examples

			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)
		

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.

A000009 counts partitions into odd parts (A066208).

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.

A160786 counts odd-length partitions of odd numbers (A300272).

A339890 counts factorizations of odd length.

A340102 counts odd-length factorizations into odd factors.

Cf. A001222, A027187, A056239, A112798, A236914, A258116, A300063, A324522, A340608, A340788, A340831.

Programs

Mathematica

Select[Range[100],OddQ[PrimeOmega[#]*PrimePi[FactorInteger[#][[-1,1]]]]&]

Formula

Intersection of A026424 (odd length) and A244991 (odd maximum).