Primitive Semiperfect Numbers which are not Primitive Abundant

Theorem

The sequence of primitive semiperfect numbers which are not also primitive abundant starts:

$6, 28, 350, 490, 496, 770, 910, 1190, \ldots$

These are semiperfect numbers which are either:

perfect

or:

whose only abundant aliquot parts are weird.

Proof

A primitive semiperfect number is a semiperfect number which has no aliquot parts which are themselves semiperfect.

Thus by definition a primitive semiperfect number is either perfect or abundant.

A primitive abundant number is an abundant number whose aliquot parts are all deficient.

Thus the perfect numbers:

$6, 28, 496, \ldots$

are not primitive abundant.

However, by Divisor of Perfect Number is Deficient, the perfect numbers are all by definition primitive semiperfect.

Hence the presence of the perfect numbers in this sequence.

Next, consider the sequence of weird numbers:

$70, 836, \ldots$

These are numbers which are abundant but not semiperfect.

Thus an abundant number whose aliquot parts are all deficient except for one or more weird numbers is a primitive semiperfect number which is not primitive abundant.

However, such a number is not primitive abundant because it has an aliquot part which is abundant, though weird.

So:

while $70$ is primitive abundant it is not primitive semiperfect number

while $350$, $490$, $710$, $910$, $1190$, and so on, are primitive semiperfect, they are not primitive abundant.

Hence the result.

$\blacksquare$