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

1,2

Comments

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

Links

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

Example

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).

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]]]]&]

Crossrefs

Complement of A300063.

Cf. A000041, A000720, A001222, A056239, A063834, A100118, A112798, A122111, A215366, A296150, A299202, A299757, A300056, A300060, A304662.

Sequence in context: A037988 A213508 A088958 * A300789 A026225 A026140

Adjacent sequences: A300058 A300059 A300060 * A300062 A300063 A300064

Keyword

nonn

Author

Gus Wiseman, Feb 23 2018

Status

approved