Freestyle perfect numbers

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The freestyle perfect numbers are the positive integers $\scriptstyle n\,$ which have some integer factorization

$n=\prod _{i=1}^{k}{f_{i}}^{e_{i}},\,$

with $\scriptstyle 1\,<\,f_{1}\,<\,\cdots \,<\,f_{k}\,\leq \,n,\,e_{i}\,>\,0\,$, such that

$2n=\prod _{i=1}^{k}{\frac {{f_{i}}^{e_{i}+1}-1}{f_{i}-1}}\,.$

This yields perfect numbers when all the $\scriptstyle f_{i}\,$ are prime factors (cf. the sum of divisors formula), and spoof perfect numbers when at least one of the $\scriptstyle f_{i}\,$ 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

$n=60=3^{1}\cdot 4^{1}\cdot 5^{1}\,$,

$\sigma _{\rm {freestyle}}(60)={\frac {3^{2}-1}{3-1}}\cdot {\frac {4^{2}-1}{4-1}}\cdot {\frac {5^{2}-1}{5-1}}=4\cdot 5\cdot 6=120=2n,\,$

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, ...}

A000396 Perfect numbers.

{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 $\emptyset \,$ = { })

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, ...?}