A300063 Heinz numbers of integer partitions of odd numbers.

2, 5, 6, 8, 11, 14, 15, 17, 18, 20, 23, 24, 26, 31, 32, 33, 35, 38, 41, 42, 44, 45, 47, 50, 51, 54, 56, 58, 59, 60, 65, 67, 68, 69, 72, 73, 74, 77, 78, 80, 83, 86, 92, 93, 95, 96, 97, 98, 99, 103, 104, 105, 106, 109, 110, 114, 119, 122, 123, 124, 125, 126, 127

Offset: 1

Author

Gus Wiseman, Feb 23 2018

Keywords

• nonn

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			15 is the Heinz number of (3,2), which has odd weight, so 15 belongs to the sequence.
Sequence of odd-weight partitions begins: (1) (3) (2,1) (1,1,1) (5) (4,1) (3,2) (7) (2,2,1) (3,1,1) (9) (2,1,1,1) (6,1).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Complement of A300061.

Cf. A000041, A000720, A001222, A056239, A063834, A112798, A215366, A296150, A299757, A300056, A300060, A304662.

Programs

Maple

a:= proc(n) option remember; local k; for k from 1+
     `if`(n=1, 0, a(n-1)) while add(numtheory[pi]
      (i[1])*i[2], i=ifactors(k)[2])::even do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 22 2018

Mathematica

Select[Range[200],OddQ[Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]]&]