A319241 Heinz numbers of strict integer partitions of even numbers. Squarefree numbers whose prime indices sum to an even number.

1, 3, 7, 10, 13, 19, 21, 22, 29, 30, 34, 37, 39, 43, 46, 53, 55, 57, 61, 62, 66, 70, 71, 79, 82, 85, 87, 89, 91, 94, 101, 102, 107, 111, 113, 115, 118, 129, 130, 131, 133, 134, 138, 139, 146, 151, 154, 155, 159, 163, 165, 166, 173, 181, 183, 186, 187, 190, 193

Offset: 1

Author

Gus Wiseman, Sep 15 2018

Keywords

• nonn

Comments

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

From Peter Munn, Feb 04 2022: (Start)

For every odd squarefree number, s, exactly one of s and 2s is a term.

Closed under the commutative operation A350066(.,.).

Closed under the commutative operation A059897(.,.) forming a subgroup of the positive integers considered as a group under A059897. As subgroups, this sequence and A028982 are each a transversal of the other.

(End)

Examples

			30 is the Heinz number of (3,2,1), which is strict and has even weight, so 30 belongs to the sequence.
The sequence of all even-weight strict partitions begins: (), (2), (4), (3,1), (6), (8), (4,2), (5,1), (10), (3,2,1), (7,1), (12), (6,2), (14), (9,1), (16), (5,3), (8,2), (18), (11,1), (5,2,1), (4,3,1).
		

Crossrefs

Complement of the union of A319242 and A013929.

Intersection of A005117 and A300061.

Cf. A000041, A000720, A001222, A008683, A056239, A296150, A300063.

Cf. A019565, A028982, A059897, A073675, A158704, A350066.

Programs

Mathematica

Select[Range[100],And[SquareFreeQ[#],EvenQ[Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]]]&]

PARI

isok(m) = issquarefree(m) && !(vecsum(apply(primepi, factor(m)[,1])) % 2); \\ Michel Marcus, Feb 08 2022

Formula

{a(n) : n >= 1} = {A019565(A158704(n)) : n >= 1} = {A073675(A319242(n)) : n >= 1}. - Peter Munn, Feb 04 2022