Definition:Primitive Semiperfect Number

Definition

A primitive semiperfect number is a semiperfect number which is not a multiple of a smaller semiperfect number.

Sequence of Primitive Semiperfect Numbers

The sequence of primitive semiperfect numbers begins:

$6, 20, 28, 88, 104, 272, 304, 350, 368, 464, 490, 496, 550, 572, \ldots$

Examples

6

$6$ is a primitive semiperfect number:

$1 + 2 + 3 = 6$

20

$20$ is a primitive semiperfect number:

$1 + 4 + 5 + 10 = 20$

28

$28$ is a primitive semiperfect number:

$1 + 2 + 4 + 7 + 14 = 28$

88

$88$ is a primitive semiperfect number:

$1 + 2 + 8 + 11 + 22 + 44 = 88$

104

$104$ is a primitive semiperfect number:

$1 + 4 + 8 + 13 + 26 + 52 = 104$

272

$272$ is a primitive semiperfect number:

$1 + 16 + 17 + 34 + 68 + 136 = 272$

and so on.

Also known as

A primitive semiperfect number is also known as:

a primitive pseudoperfect number

an irreducible pseudoperfect number

an irreducible semiperfect number

Semiperfect can also be seen hyphenated: semi-perfect.

Also see

• Multiple of Semiperfect Number is Semiperfect

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $104$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $104$

• Weisstein, Eric W. "Primitive Pseudoperfect Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PrimitivePseudoperfectNumber.html