A340931 Heinz numbers of integer partitions of odd numbers into an odd number of parts.

2, 5, 8, 11, 17, 18, 20, 23, 31, 32, 41, 42, 44, 45, 47, 50, 59, 67, 68, 72, 73, 78, 80, 83, 92, 97, 98, 99, 103, 105, 109, 110, 114, 124, 125, 127, 128, 137, 149, 153, 157, 162, 164, 167, 168, 170, 174, 176, 179, 180, 182, 188, 191, 195, 197, 200, 207, 211

Offset

1,1

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This is a bijective correspondence between positive integers and integer partitions.

Links

Table of n, a(n) for n=1..58.

Formula

Intersection of A026424 and A300063.

Example

The sequence of terms together with the corresponding partitions begins:
      2: (1)             50: (3,3,1)        109: (29)
      5: (3)             59: (17)           110: (5,3,1)
      8: (1,1,1)         67: (19)           114: (8,2,1)
     11: (5)             68: (7,1,1)        124: (11,1,1)
     17: (7)             72: (2,2,1,1,1)    125: (3,3,3)
     18: (2,2,1)         73: (21)           127: (31)
     20: (3,1,1)         78: (6,2,1)        128: (1,1,1,1,1,1,1)
     23: (9)             80: (3,1,1,1,1)    137: (33)
     31: (11)            83: (23)           149: (35)
     32: (1,1,1,1,1)     92: (9,1,1)        153: (7,2,2)
     41: (13)            97: (25)           157: (37)
     42: (4,2,1)         98: (4,4,1)        162: (2,2,2,2,1)
     44: (5,1,1)         99: (5,2,2)        164: (13,1,1)
     45: (3,2,2)        103: (27)           167: (39)
     47: (15)           105: (4,3,2)        168: (4,2,1,1,1)

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], OddQ[PrimeOmega[#]]&&OddQ[Total[primeMS[#]]]&]

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.

These partitions are counted by A160786.

The even version is A236913 (A340784).

The case of where the prime indices are also odd is A300272.

A000009 counts partitions into odd parts (A066208).

A001222 counts prime factors.

A027193 counts odd-length partitions (A026424).

A047993 counts balanced partitions (A106529).

A056239 adds up prime indices.

A058695 counts partitions of odd numbers (A300063).

A072233 counts partitions by sum and length.

A112798 lists the prime indices of each positive integer.

Cf. A000041, A316413, A326845, A340385, A340386, A340387, A340604, A340607, A340854, A340855.

Sequence in context: A186496 A032765 A375298 * A261416 A340386 A300272

Adjacent sequences: A340928 A340929 A340930 * A340932 A340933 A340934

Keyword

nonn

Author

Gus Wiseman, Feb 05 2021

Status

approved