A300061 Heinz numbers of integer partitions of even numbers.

1, 3, 4, 7, 9, 10, 12, 13, 16, 19, 21, 22, 25, 27, 28, 29, 30, 34, 36, 37, 39, 40, 43, 46, 48, 49, 52, 53, 55, 57, 61, 62, 63, 64, 66, 70, 71, 75, 76, 79, 81, 82, 84, 85, 87, 88, 89, 90, 91, 94, 100, 101, 102, 107, 108, 111, 112, 113, 115, 116, 117, 118, 120

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

			75 is the Heinz number of (3,3,2), which has even weight, so 75 belongs to the sequence.
Sequence of even-weight partitions begins: () (2) (1,1) (4) (2,2) (3,1) (2,1,1) (6) (1,1,1,1) (8) (4,2) (5,1) (3,3) (2,2,2) (4,1,1).
		

Links

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

Crossrefs

Complement of A300063.

Cf. A000041, A000720, A001222, A056239, A063834, A100118, A112798, A122111, A215366, A296150, A299202, 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])::odd do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 22 2018

Mathematica

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