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For all n>1, sigma(n)>n, therefore 2n/sigma(n)-1 is always less than 1, i.e., k>0.

For k=1 to 11, the smallest known numbers to give 1/2^k are: 3, 15, 135, 2295, 1485, 1365, 63855, 16410735, 397575, 667275, 271543725.

For k=12 to 19, they are: unknown, 741585912975, unknown, 39206559148911, 2569480266942180207, 1712973775775070501, unknown, 299364435975778645966263.

Spoof-perfect numbers are freestyle perfect numbers which are not perfect numbers.

Only one odd spoof perfect number is known: 198585576189, found by Descartes.

Assuming all integer factorizations were tried in the range [1..9900] in A058007, where I removed 6, 28, 496, 8128 from the list (I did not do the search for spoof perfect numbers myself, so the accuracy of my list depends on the accuracy of A058007's list.)

Roughly said, a spoof-perfect number is a number that would be perfect if some (one or more) of its composite factors were wrongly assumed to be prime, i.e., taken as a spoof prime.

Contribution from M. F. Hasler, Jan 13 2013: (Start)

I added "roughly said" to the above last phrase, since different interpretations of "would be perfect if some of its composite factors were wrongly assumed to be prime" are possible, and Descartes's example does not help to decide: (Notations are those from A058007, n = Sum (f_i)^(e_i).)

(a) If a spoof prime factor f_i is composite, may it have some of the smaller (spoof or true prime) f_j as factors or not? (In Descartes's example, this is not the case. And "assumed to be prime" could well imply that the answer is "no". But there is no such restriction in A058007.)

(b) If f_i is composite, is it required that e_i is the highest possible power, i.e., the (f_i)-valuation of n (or of n divided by all smaller f_j to the powers e_j)? (In Descartes's example this is the case. And if product(f_i^e_i) is to be a "prime factorization" of n, then it should be the case. But there is no such restriction in A058007. Note that this is not a consequence of (a), because the f_i could have common factors: e.g., even if f_1=21, f_2=35, f_3=45 are "wrongly assumed to be prime", then n=21*35*45 would have the (f_1)-valuation = 2, i.e., factorization n = f_1^2*75.)

(c) Is it reasonable to allow for even spoof primes f_i? (In Descartes's example this is not the case. And it seems somehow inconceivable that an even number be "wrongly assumed to be prime". But there is no such restriction in A058007.

Depending on the answer to each of these questions, "spoof-perfect numbers" as defined using "composite factors were wrongly assumed to be prime", could mean at least 8 different sequences. (End)

The number 198585576189 = 9018009*22021 is the first and only known odd term in A058007 and A174292; 9018009 = A222263(79).

If the number x is a prime which does not divide n, then n*x is a perfect number. This happens (so far) only when x = 2n-1 = 2^p-1 is a Mersenne prime (cf. A000043). But if x does not divide n, as, e.g., for (n,x)=(10,9), then n*x is a so-called freestyle perfect number, cf. A058007: Namely it "would be perfect if x is assumed to be prime", which means that sigma(n*x) is replaced by sigma(n)*(x+1) in the relation 2P=sigma(P) characterizing perfect numbers P, listed in A000396.

See also the (more interesting) subsequence of odd terms, A222263.

This list of 7 numbers is conjectured to be complete.

They are mentioned in conjunction with an example of a family of 7 isotopic amicable pairs (see Example 2.4 on page 3 of the Garcia et al. link).

Start from the amicable pair (961388889147024075, 961453639626095925). It can be written as (g*m, g*n), with g=3^3*5^2*19*31=397575, with sigma(g)/g=1024/513 and GCD(g,m) = GCD(g,n) = 1. Then by replacing g any of the other 6 values of the sequence, we obtain 6 other amicable pairs, isotopic to the first one.

These numbers belong to A222263 with k = 1/512.