Primitive Weird Numbers Overview

• Primitive weird numbers are abundant integers that lack a semiperfect subset and have all proper divisors deficient.
• Researchers develop algorithmic constructions using centers and index sequences to generate and enumerate these numbers.
• Computational studies reveal rapid complexity growth and highlight open problems regarding odd cases and high prime exponents.

A primitive weird number is a positive integer that is both weird and possesses the additional property that none of its proper divisors are themselves weird. These numbers are situated at the intersection of divisor function theory, additive combinatorics, and computational number theory, and exhibit rich structure related to their prime factorization and abundance. Research has progressively expanded the list of known primitive weird numbers and clarified sufficient and necessary conditions for their construction, as well as algorithmic methods for their discovery (Amato et al., 2018, Amato et al., 2018, Iannucci, 2015).

1. Formal Definitions and Structural Properties

Let $n\in\mathbb{N}$. The sum-of-divisors function is defined by

$\sigma(n) = \sum_{d \mid n} d.$

Define abundance and deficiency as

$$A(n) = \sigma(n) - 2n, \qquad d(n) = 2n - \sigma(n) = -A(n).$$

• $n$ is deficient if $\sigma(n)<2n$ ($d(n)>0$).
• $n$ is perfect if $\sigma(n)=2n$ ($A(n)=0$).
• $n$ is abundant if $\sigma(n) = \sum_{d \mid n} d.$0 ($\sigma(n) = \sum_{d \mid n} d.$1).

A number $\sigma(n) = \sum_{d \mid n} d.$2 is semiperfect (or pseudoperfect) if it is abundant and some subset of its proper divisors sums to $\sigma(n) = \sum_{d \mid n} d.$3. Weird numbers are abundant numbers which are not semiperfect—no such subset exists.

A primitive weird number is a weird number none of whose proper divisors is weird (Amato et al., 2018, Amato et al., 2018). Equivalently, $\sigma(n) = \sum_{d \mid n} d.$4 is primitive weird when it is both weird and primitive abundant—that is, all its proper divisors are deficient.

The sum $\sigma(n) = \sum_{d \mid n} d.$5 is the total number of prime factors of $\sigma(n) = \sum_{d \mid n} d.$6, counted with multiplicity.

2. Generation and Sufficient Conditions

Primitive weird numbers can be algorithmically constructed by analyzing two related classes: primitive abundant numbers (PANs) and weird numbers.

• A number $\sigma(n) = \sum_{d \mid n} d.$7 is a PAN if $\sigma(n) = \sum_{d \mid n} d.$8 and every proper divisor $\sigma(n) = \sum_{d \mid n} d.$9 satisfies $$A(n) = \sigma(n) - 2n, \qquad d(n) = 2n - \sigma(n) = -A(n).$$0.
• Primitive weird numbers (PWNs) are exactly those PANs that are also weird.

A key innovation is the introduction of the center of a deficient number,

$$A(n) = \sigma(n) - 2n, \qquad d(n) = 2n - \sigma(n) = -A(n).$$1

for deficient $$A(n) = \sigma(n) - 2n, \qquad d(n) = 2n - \sigma(n) = -A(n).$$2. For $$A(n) = \sigma(n) - 2n, \qquad d(n) = 2n - \sigma(n) = -A(n).$$3 deficient and $$A(n) = \sigma(n) - 2n, \qquad d(n) = 2n - \sigma(n) = -A(n).$$4,

• $$A(n) = \sigma(n) - 2n, \qquad d(n) = 2n - \sigma(n) = -A(n).$$5 is abundant iff $$A(n) = \sigma(n) - 2n, \qquad d(n) = 2n - \sigma(n) = -A(n).$$6.
• $$A(n) = \sigma(n) - 2n, \qquad d(n) = 2n - \sigma(n) = -A(n).$$7 is deficient iff $$A(n) = \sigma(n) - 2n, \qquad d(n) = 2n - \sigma(n) = -A(n).$$8.

Theorem 3.1 (Amato et al., 2018) provides constructive sufficient conditions:

• Given $$A(n) = \sigma(n) - 2n, \qquad d(n) = 2n - \sigma(n) = -A(n).$$9 deficient, select primes $n$0 with $n$1 and let $n$2.
• If $n$3 is abundant and $n$4 falls in a precisely defined interval (the set $n$5 based on the "gap" structure of the prime choices), then $n$6 is primitive weird.

An alternative approach is specialized for numbers of the form $n$7, where $n$8 are odd primes. For these, necessary and sufficient conditions for weirdness and primitivity are explicitly characterized (Iannucci, 2015):

• $n$9, $\sigma(n)<2n$0, and $\sigma(n)<2n$1 for $\sigma(n)<2n$2.
• $\sigma(n)<2n$3 and $\sigma(n)<2n$4 must be in the interval $\sigma(n)<2n$5 and prime.
• There is no subset of proper divisors summing to $\sigma(n)<2n$6. For such $\sigma(n)<2n$7, weirdness implies primitivity.

3. Algorithmic Enumeration and Computational Results

Modern enumeration employs recursive construction using centers, with efficient pruning based on whether partial products can ever yield abundance (deficient sequence completion). Separate algorithms exist for square-free and general cases.

PWNs with a fixed number $\sigma(n)<2n$8 of prime factors are enumerated by recursively building up from 1, selecting primes in appropriate intervals dictated by the center $\sigma(n)<2n$9 at each step, and testing for abundance and (finally) weirdness via subset-sum checks (Amato et al., 2018).

For numbers with prime powers, further care is taken to test primitivity (no non-deficient proper divisor) using criteria derived from the arithmetic structure.

Extensive computational searches have produced the following enumeration results:

• All PWNs with up to $d(n)>0$0 prime factors have been constructed, a significant extension from previous bounds ($d(n)>0$1) (Amato et al., 2018).
• The largest PWN known has $d(n)>0$2 distinct prime factors and $d(n)>0$3 digits; the previous record was $d(n)>0$4 digits.
• For $d(n)>0$5 primitive weird numbers, all such numbers for $d(n)>0$6 are tabulated, comprising thousands of distinct examples (Iannucci, 2015).

4. Distributional Features and Factorization Patterns

All known primitive weird numbers are even. It is open whether any odd weird number exists (the question is current as of the surveyed research and known as Erdős’s $d(n)>0$7).

Previous to recent work, virtually all PWNs described had square-free odd parts; only five with an odd prime squared were known (Amato et al., 2018). Advances include:

• No PWN containing an odd prime square exists for $d(n)>0$8, and none with two squares or a cube for $d(n)>0$9.
• First examples with two distinct odd primes squared appear at $n$0; a single example with three squares at $n$1.
• No PWNs with any odd prime to exponent $n$2 for $n$3 have been found.

A summary of these existence and nonexistence findings is given in Theorem 4.7 of (Amato et al., 2018).

5. Illustrative Examples

• For $n$4, the smallest PWN is $n$5 with $n$6. It is weird, and all proper divisors are deficient.
• Large PWN examples with high $n$7 are described by their prime lists and index sequences (relative position with respect to $n$8 at each step). A PWN with $n$9 is $\sigma(n)=2n$0 (see Table 4 in (Amato et al., 2018)).
• For $\sigma(n)=2n$1, complete tables exist for $\sigma(n)=2n$2 through $\sigma(n)=2n$3; e.g., $\sigma(n)=2n$4 ($\sigma(n)=2n$5), $\sigma(n)=2n$6 ($\sigma(n)=2n$7), and others (Iannucci, 2015).

6. Open Problems and Conjectures

Several fundamental questions remain unresolved:

• The infinitude of primitive weird numbers is conjectured but unproved (Iannucci, 2015, Amato et al., 2018).
• No primitive weird number with a cube or higher power of an odd prime factor is known.
• No odd weird number has been found up to $\sigma(n)=2n$8.
• The boundedness of the ratio $\sigma(n)=2n$9 on all weird numbers (not necessarily primitive) is unknown.
• It is empirically observed that counts of $A(n)=0$0 primitive weirds grow rapidly with $A(n)=0$1, but proof of unboundedness is lacking.

Current algorithms employing the center $A(n)=0$2 and index sequences enable continued exploration for PWNs with even larger $A(n)=0$3, and all code for these searches is publicly available (Amato et al., 2018).

7. Summary Table of PWN Existence for Small $A(n)=0$4

$A(n)=0$5	Known PWN with odd prime square?	Max known number of PWNs	Largest example (digits)
$A(n)=0$66	No	$A(n)=0$7	$A(n)=0$820–30 digits
7	Yes (single square only)	$A(n)=0$9
8–11	Yes (hundreds, single square)
12	Yes (two squares, several)
15	Yes (three squares, one)		$n$0
$n$1	Unknown	—	—

The absence of PWNs with higher odd exponents and the open status of odd weirds suggest deep underlying structure yet to be fully uncovered (Amato et al., 2018).