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# Freestyle perfect numbers

The freestyle perfect numbers are the positive integers which have some integer factorization

with , such that

This yields perfect numbers when all the are prime factors (cf. the sum of divisors formula), and spoof perfect numbers when at least one of the is a spoof-prime (i.e., composite) factor, in which case the preceding formula can be seen as an erroneous or inappropriate use of the sum of divisors formula.

## Examples

- ,

so 60 is a freestyle perfect number (although a spoof perfect number).

## Sequences

A058007 Freestyle perfect numbers.

- {6, 28, 60, 84, 90, 120, 336, 496, 840, 924, 1008, 1080, 1260, 1320, 1440, 1680, 1980, 2016, 2160, 2184, 2520, 2772, 3024, 3420, 3600, 3780, 4680, 5040, 5940, 6048, 6552, 7440, 7560, 7800, 8128, 8190, ...}

- {6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216, ...}

A?????? Even perfect numbers. (A000396 only if no odd perfect numbers exist, this is an open problem!)

- {6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216, ...}

A?????? Odd perfect numbers. (A000000, i.e. the empty sequence, unless they do exist, this is an open problem!)

- {?} (or = { })

A174292 Spoof perfect numbers: 198585576189 is Descartes' number, the only odd spoof perfect number ever found! *Assuming all integer factorizations where tried in the range [1..9900] in A058007, where I removed the perfect numbers 6, 28, 496, 8128 from the list, they are...*

- {60, 84, 90, 120, 336, 840, 924, 1008, 1080, 1260, 1320, 1440, 1680, 1980, 2016, 2160, 2184, 2520, 2772, 3024, 3420, 3600, 3780, 4680, 5040, 5940, 6048, 6552, 7440, 7560, 7800, 8190, ..., 198585576189, ...}

A?????? Even spoof perfect numbers: *Assuming all integer factorizations where tried in the range [1..9900] in A058007, where I removed the perfect numbers 6, 28, 496, 8128 from the list, they are...*

- {60, 84, 90, 120, 336, 840, 924, 1008, 1080, 1260, 1320, 1440, 1680, 1980, 2016, 2160, 2184, 2520, 2772, 3024, 3420, 3600, 3780, 4680, 5040, 5940, 6048, 6552, 7440, 7560, 7800, 8190, ...}

A?????? Odd spoof perfect numbers: 198585576189 is Descartes' number, the only odd spoof perfect number ever found!

- {198585576189, ...?}