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Spoof perfect numbers

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A spoof perfect number is a positive integer that would be a perfect number if some of its [positive] composite factors were treated as if they were spoof-prime factors in the sum of divisors formula.

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, ...}

The spoof perfect numbers are freestyle perfect numbers which are not perfect numbers, i.e., for which some integer factorization

where ; may differ from (the number of distinct prime factors of ), and such that

when at least one of the is incorrectly assumed to be prime.

Example:

,

so 60 is a spoof perfect number.

Spoof-prime factors

As spoof-prime factors (spoof-prime divisors) of , one might consider:

  • any odd composite unitary divisors , , of as odd spoof-prime unitary divisors (odd spoof-prime unitary factors) of
  • any even/odd composite unitary divisors , , of as spoof-prime unitary divisors (spoof-prime unitary factors) of
  • any positive odd composite integer as spoof-prime factor
  • any positive even/odd composite integer as spoof-prime factor
  • any negative/positive odd composite integer as spoof-prime factor
  • any negative/positive even/odd composite integer as spoof-prime factor
  • any negative/positive odd composite integer and/or negated odd prime as spoof-prime factor[1]
  • any negative/positive even/odd composite integer and/or negated prime as spoof-prime factor

Quasi-prime factors

Quasi-prime number: A positive integer without "small" prime factors.[2]

This means that all prime factors of must be greater than , where is a function that increases more slowly than . For example,

.

Now "quasi-prime factors" brings many more possible definitions of spoof-perfect numbers! (It depends on the choice of .) These would make other sequences of (spoof-perfect numbers, maybe a modified name) too!

Even spoof perfect numbers

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, ...}

Odd spoof perfect numbers

Main article page: Odd spoof perfect numbers

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

{198585576189, ...?}

See also

Notes

  1. Greg Martin also considered negated primes as spoof-primes. See: Richard K. Guy, Unsolved Problems in Number Theory (2004), p. 72.
  2. Quasi-prime number. B.M. Bredikhin (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Quasi-prime_number&oldid=19152

References

  • Richard K. Guy, Unsolved Problems in Number Theory (2004), p. 72.