Poppins numbers and A229088

[Allan Wechsler]

The multiperfect numbers are those n such that n divides sigma(n) (the sum of divisors of n). That is, when the fraction sigma(n)/n is reduced to lowest terms, the denominator being 1 means n is multiperfect.

The denominator of sigma(n)/n is at oeis.org/A017666 .

Some numbers just fail to be multiperfect, in the sense that when you reduce sigma(n)/n to lowest terms, almost everything cancels. An example is:

L = 951870502263811526107776159523473265965115414366224136036102963200

= 2^32 3^13 5^2 7^6 11^2 17 19 23 29 31 41 89 163 263 307 547^2 613 1093 2141 4733 599479

In this case, sigma(L)/L = 25/4. I have been jokingly thinking of numbers like this as "Poppins numbers", remembering that Mary Poppins thought of herself a "practically perfect in every way". I don't have a strong definition of these -- that is, I have no clear criterion for how small the denominator has to be before we take notice of the number as a near miss in the search for multi-perfection.

OEIS, though, thinks that if the denominator is 6 or less, that's interesting.

Numbers for which this denominator (let's call it D) is 1 are of course the multiperfect numbers themselves, oeis.org/A007691 .

Numbers for which D = 2 are striking enough that they actually have a name, "hemiperfect" numbers. They are at oeis.org/A159907 .

D = 3 doesn't have a name, but I propose calling them "tritoperfect" on analogy with "hemiperfect". See oeis.org/A245775 .

D = 4, 5, and 6 are at oeis.org/A229088 , oeis.org/A067237 , and oeis.org/A262356, respectively. We could call them tetartoperfect, pemptoperfect, and hektoperfect numbers (from the Greek words for "third", "fourth", and "fifth").

BUT: The identity of A229088 with the tetartoperfect numbers is only conjectural. This sequence was added to OEIS with a different original interpretation. The title interpretation is that these are the numbers k, such that sigma(k) and antisigma(k) (the sum of the nondivisors of k less than k) have the same residue modulo k. 

Now the punchline: my clumsy calculations suggest that the number L given above is a counterexample to the conjecture. I am pretty sure that this number is tetartoperfect, and I am also pretty sure that it doesn't belong in A229088, because the residue mod L of its antisigma is exactly twice the sigma residue, not the same number.

Can somebody confirm this? I'm feeling all thumbs and I half suspect I have made a stupid arithmetic error.

Also: if anybody thinks the D = 7 "hebdomoperfect" numbers are interesting, the first few are 7, 14, 42, 56, 168, 280, 588, 840, 2520.

Everybody who searches for multiperfect numbers stumbles on Poppinses like L from time to time, and it would be nice to have a place to record them.

-- Allan

[M F Hasler]

On Tue, May 26, 2026 at 4:45 PM Allan Wechsler <...> wrote:

(...) Some numbers just fail to be multiperfect, in the sense that when you reduce sigma(n)/n to lowest terms, almost everything cancels.(...)

You might also be interested in 

A174292 = Spoof-perfect numbers: Freestyle perfect numbers (A058007) which are not perfect numbers (A000396).

also discussed on  OEIS wiki: Spoof perfect numbers.

Descartes found one odd Spoof Perfect Number, N = 3² ⋅  7² ⋅  11² ⋅  13² ⋅  22021,

which would be perfect, i.e.,  σ(N)  =  σ(3²) ⋅ σ(7²) ⋅ σ(11²) ⋅ σ(13²) ⋅ σ(22021)  =?=  2 N,

if 22021 was prime <=>  σ(22021)  =?=  1 + 22021  (but actually  22021  = 19²  ⋅  61).

To me, the most interesting challenge is to find other *odd* freestyle perfect numbers,

question first asked by John Leech according to B1, pp. 44-45, UPiNT2, R.K.Guy

[but obviously already Descartes implicitly asked that question...]

and also subject of Carlos Rivera's primepuzzles.net/puzzles/puzz_111.htm.

About 13 years ago I have added some considerations in A174292, which actually concern the definition of the freestyle perfect numbers A058007 -- namely, what are reasonable conditions for mistakenly considering a composite factor prime: It seems quite natural to exclude the possibility that a very small prime be a divisor; in particular:

- the "spoof prime" factor should (IMHO quite obviously) not be even;

- none of the smaller prime factors of the number should divide the "spoof prime"(composite) factor:

  If you know that 3 divides N, you would naturally factor out all factors of 3.

- maybe the "spoof prime" factors should necessarily be the largest factor(s)

- no two factors should have a gcd > 1 (because testing primality may be considered nontrivial, but the Euclidean algorithm is straightforwardly applied to a pair of even huge numbers, to reveal very quickly a gcd.

Descartes' 198585576189 satisfies all of these criteria, but  none of them is required in the definition of A058007; yet no other odd term is known!

- Maximilian