Melfi 2015 conditional infiniteness primitive weird numbers

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theorem_1: For primes p = 2^(k+2) - a and q = 2^(k+2) + b with a, b odd and b + 3 < a < 2^((k-1)/2), the number 2^k p q is primitive weird; with the paper's conditional deduction of infinitely many primitive weird numbers from a prime-gap bound.

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G. Melfi, On the conditional infiniteness of primitive weird numbers, J. Number Theory 147 (2015), 508--514 (received 30 January 2014, revised 8 July 2014, accepted 28 August 2014, available online 16 September 2014); DOI 10.1016/j.jnt.2014.07.024; MSC 11A25, 11B83. The same author's 2004 survey is filed as melfi_2004_certain_positive_integer_sequences.

The copy read for this card is the publisher's PDF (Elsevier), seven physical pages, printed pp. 508--514 (PDF p. $n$ is printed p. $507+n$), typeset with a clean text layer. Provenance: obtained in September 2026; the PDF names the DOI address http://dx.doi.org/10.1016/j.jnt.2014.07.024; 245,588 bytes. Read status: claims checked for Theorem 1 and the conditional infinitude statement (pp. 509--510), read in the text layer; the proof of Theorem 1 (section 3, pp. 510--512) was followed for structure only and not verified. That copy prints "0022-314X/© 2014 Elsevier Inc. All rights reserved." on its first page, every other right reserved.

Contents

$\sigma(n)$ is the sum of divisors of $n$; $n$ is abundant if $\sigma(n)>2n$, semiperfect (pseudoperfect) if it is a sum of distinct proper divisors, weird if abundant and not semiperfect, and primitive weird if weird and not a multiple of another weird number; $\Delta(n)=\sigma(n)-2n$ is the abundance.

• Background (pp. 508--509): the term "weird" is Benkoski's (1972); Benkoski--Erdős [4] proved that there are infinitely many weird numbers, of positive asymptotic density; if $n$ is weird and $p>\sigma(n)$ is prime then $np$ is weird ([6]; Lemma 3, p. 510), which motivates the primitive ones. Whether infinitely many primitive weird numbers exist was posed by Benkoski and Erdős as a question, still open in [8, p. 77] and [12, p. 43]; a list of those not exceeding $1.8\cdot10^9$ is OEIS A002975; Kravitz (1976) proved that for a prime $p>2^k$, if $q=(2^kp-(p+1))/((p+1)-2^k)$ is prime then $2^{k-1}pq$ is primitive weird, and found eleven weird numbers, among them a 53-digit one that long held the primitive record; Klyve (2013) announced a 226-digit weird number.
• Theorem 1 (p. 509; proof pp. 510--512): let $k$ be a positive integer and $a$, $b$ positive odd integers such that $p=2^{k+2}-a$ and $q=2^{k+2}+b$ are primes. If $b+3<a<2^{(k-1)/2}$ then $n=2^kpq$ is a primitive weird number. The least triple is $(a,b,k)=(5,1,6)$, giving $2^6(2^8-5)(2^8+1)=4\,128\,448$, the 32nd primitive weird number; there is no other triple with $k\le7$; 116 of the first 160 primitive weird numbers have the form $2^kpq$. The proof shows $n$ abundant ($\Delta(n)=2^{k+1}(a-b-3)+(a-1)(b+1)>0$), primitive abundant ($\Delta(n/p),\Delta(n/q),\Delta(n/2)<0$), and weird by Lemma 2 (p. 510: an abundant $n$ is weird iff $\Delta(n)$ is not a sum of distinct proper divisors), locating $\Delta(n)$ in a gap between the intervals $I_h$ that contain every sum of distinct proper divisors of $n$ up to $2^{3k/2}$.
• Conditional infinitude (pp. 509--510): if $p_{n+1}-p_n<0.1\,p_n^{1/2}$ for all sufficiently large $n$, Theorem 1 yields infinitely many primitive weird numbers of the form $2^kpq$; Cramér's conjecture, or the weaker Gonek conjecture that $p_{n+1}-p_n<p_n^\epsilon$ for every $\epsilon>0$ and large $n$, suffices, and the unconditional Baker--Harman--Pintz bound $p_n^{0.525}$ is "very close" to what is needed.
• Section 4 (pp. 512--513): Conjecture 1, infinitely many primitive weird numbers of the form $2^kpq$; examples from PARI such as $k=5898$, $a=4529$, $b=4171$, a 5328-digit primitive weird number; Conjecture 2, $\liminf(w_{n+1}-w_n)/w_n=0$ for the sequence of primitive weird numbers, while the absence of primitive weird numbers between $1.74\cdot10^8$ and $2.54\cdot10^8$ leaves a positive $\limsup$ possible; the remark (p. 513) that the proof of Theorem 1 "can be easily adapted" when $2^k$ is replaced by an almost perfect number $m$ ($\sigma(m)=2m-1$), an adaptation not written out: if $p=4m-a$ and $q=4m+b$ are primes with odd positive $a,b$ and $b+3<a<\sqrt{m/2}$ then $mpq$ is primitive weird, so an odd almost perfect number above $1$, with suitable primes $p,q$, would give an odd weird number.

Compiled scope

The whole paper (pp. 508--514) was read in the text layer; the proof of Theorem 1 was followed for structure but not checked line by line. Nothing here is independently reviewed.

Bears on. #470: Theorem 1 constructs primitive weird numbers $2^kpq$ from prime pairs near $2^{k+2}$, and the paper deduces from it (pp. 509--510) that the problem's second question (infinitely many primitive weird numbers) has a yes answer under the unproved prime-gap bound $p_{n+1}-p_n<0.1\,p_n^{1/2}$ for large $n$; the odd-weird question is touched only by the remark that an odd almost perfect number above $1$, with suitable primes $p,q$, would yield an odd weird number.

Result. Theorem 1, with the conditional consequence and the almost perfect remark.

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