--- title: Primitive Weird Numbers Overview url: https://www.emergentmind.com/topics/primitive-weird-numbers type: topic --- # Primitive Weird Numbers Overview 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 [1802.07178, 1803.00324, 1504.02761]. ## 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 $\sigma(n)>2n$ ($A(n)>0$). A number $n$ is *semiperfect* (or *pseudoperfect*) if it is abundant and some subset of its proper divisors sums to $n$. *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 [1802.07178, 1803.00324]. Equivalently, $n$ is primitive weird when it is both weird and primitive abundant—that is, all its proper divisors are deficient. The sum $\Omega(n) = \sum_{p^\alpha \Vert n} \alpha$ is the total number of prime factors of $n$, 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 $m$ is a PAN if $\sigma(m) > 2m$ and every proper divisor $d \mid m$ satisfies $\sigma(d) < 2d$. - 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, \[ c(m) = \frac{\sigma(m)}{\sigma(m) - 2m} = \frac{\sigma(m)}{d(m)} \] for deficient $m$. For $m$ deficient and $(m,p)=1$, - $mp$ is abundant iff $p < c(m)$. - $mp$ is deficient iff $p > c(m)$. Theorem 3.1 [1803.00324] provides constructive sufficient conditions: - Given $m$ deficient, select primes $p_1,\dots,p_k$ with $\sigma(m)+1 < p_1 < \cdots < p_k$ and let $w = m p_1\cdots p_k$. - If $w$ is abundant and $A(w)$ falls in a precisely defined interval (the set $U_{m,p_1,p_k}$ based on the "gap" structure of the prime choices), then $w$ is primitive weird. An alternative approach is specialized for numbers of the form $2^k p q$, where $p 0$, $4 \mid a(n)$, and $M < a(n) < M(M+1)$ for $M = 2^{k+1} - 1$. - $p$ and $q$ must be in the interval $(M, 2M)$ and prime. - There is no subset of proper divisors summing to $n$. For such $n$, 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 $\Omega$ of prime factors are enumerated by recursively building up from 1, selecting primes in appropriate intervals dictated by the center $c(m)$ at each step, and testing for abundance and (finally) weirdness via subset-sum checks [1802.07178]. 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 $\Omega=16$ prime factors have been constructed, a significant extension from previous bounds ($\Omega \leq 6$) [1802.07178]. - The largest PWN known has $16$ distinct prime factors and $14\,712$ digits; the previous record was $5\,328$ digits. - For $2^k p q$ primitive weird numbers, all such numbers for $1 \leq k \leq 14$ are tabulated, comprising thousands of distinct examples [1504.02761]. ## 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 $25 problem$). Previous to recent work, virtually all PWNs described had square-free odd parts; only five with an odd prime squared were known [1802.07178]. Advances include: - No PWN containing an odd prime square exists for $\Omega < 7$, and none with two squares or a cube for $\Omega=7$. - First examples with two distinct odd primes squared appear at $\Omega=12$; a single example with three squares at $\Omega=15$. - No PWNs with any odd prime to exponent $\geq 3$ for $\Omega \leq 16$ have been found. A summary of these existence and nonexistence findings is given in Theorem 4.7 of [1802.07178]. ## 5. Illustrative Examples - For $\Omega=4$, the smallest PWN is $770 = 2 \cdot 5 \cdot 7 \cdot 11$ with $A(770)=192$. It is weird, and all proper divisors are deficient. - Large PWN examples with high $\Omega$ are described by their prime lists and index sequences (relative position with respect to $c(m)$ at each step). A PWN with $\Omega=10$ is $2 \cdot 5 \cdot 11 \cdot 127 \cdot 167 \cdot 223 \cdot 311 \cdot 1327 \cdot 2011 \cdot 2801$ (see Table 4 in [1802.07178]). - For $n=2^k p q$, complete tables exist for $k=1$ through $k=14$; e.g., $70 = 2 \cdot 5 \cdot 7$ ($k=1$), $9272 = 2^3 \cdot 19 \cdot 61$ ($k=3$), and others [1504.02761]. ## 6. Open Problems and Conjectures Several fundamental questions remain unresolved: - The infinitude of primitive weird numbers is conjectured but unproved [1504.02761, 1802.07178]. - 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 $10^{21}$. - The boundedness of the ratio $\sigma(n)/n$ on all weird numbers (not necessarily primitive) is unknown. - It is empirically observed that counts of $2^k p q$ primitive weirds grow rapidly with $k$, but proof of unboundedness is lacking. Current algorithms employing the center $c(m)$ and index sequences enable continued exploration for PWNs with even larger $\Omega$, and all code for these searches is publicly available [1802.07178]. ## 7. Summary Table of PWN Existence for Small $\Omega$ | $\Omega$ | Known PWN with odd prime square? | Max known number of PWNs | Largest example (digits) | |----------|----------------------------------|-------------------------|--------------------------| | $\leq$6 | No | $1001$ | $\sim$20–30 digits | | 7 | Yes (single square only) | $8192$ | | | 8–11 | Yes (hundreds, single square) | | | | 12 | Yes (two squares, several) | | | | 15 | Yes (three squares, one) | | $14\,712$ | | $>16$ | 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 [1802.07178]. Source: https://www.emergentmind.com/topics/primitive-weird-numbers