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

1,2

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)

Links

Table of n, a(n) for n=1..59.

Formula

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

Example

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).

Mathematica

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

Prog

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

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.

Sequence in context: A255607 A310185 A339620 * A310186 A363022 A289167

Adjacent sequences: A319238 A319239 A319240 * A319242 A319243 A319244

Keyword

nonn

Author

Gus Wiseman, Sep 15 2018

Status

approved