Port Fillings for Primary Pseudoperfect Numbers

Han Wang

arxiv: 2605.21518 · v1 · pith:EFX5MRZVnew · submitted 2026-05-18 · 🧮 math.NT

Port Fillings for Primary Pseudoperfect Numbers

Han Wang This is my paper

Pith reviewed 2026-05-22 00:47 UTC · model grok-4.3

classification 🧮 math.NT

keywords primary pseudoperfect numbersport fillingsarithmetic derivativecomposition lawErdős problemsquarefree integersresidual equationsreciprocal sums

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The pith

The product rule for the arithmetic derivative provides a composition law for ports that separates inherited fillings from primitive ones for primary pseudoperfect numbers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a framework of ports and fillings to study primary pseudoperfect numbers, squarefree integers n satisfying 1/n plus the sum of 1/p over prime divisors p of n equals 1. This condition is equivalent to Erdős's question on whether there are infinitely many finite sets of distinct primes whose reciprocals sum to 1 minus 1/m. The work applies the product rule of the arithmetic derivative to obtain a composition law for ports, defined as pairs (R, c) where a squarefree B fills the port if cB minus R times the arithmetic derivative of B equals 1. This law distinguishes fillings inherited from smaller primary pseudoperfect numbers from primitive fillings relative to a fixed residual equation, offering a structured approach to classify and generate such numbers.

Core claim

The product rule for the arithmetic derivative gives the composition law for ports. This law separates fillings inherited from smaller primary pseudoperfect numbers from fillings that are primitive relative to the fixed residual equation.

What carries the argument

The port composition law derived from the product rule of the arithmetic derivative, which separates inherited fillings from primitive fillings relative to a residual equation.

If this is right

Primary pseudoperfect numbers arise as port fillings of residual equations.
The composition law allows construction of new fillings by combining smaller primary pseudoperfect numbers.
Primitive fillings relative to a fixed residual equation yield primary pseudoperfect numbers not derived from smaller instances.
This separation supports analysis of primary pseudoperfect numbers with larger numbers of prime factors.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Existence of primitive fillings for arbitrarily large residual equations would imply infinitely many primary pseudoperfect numbers.
The port framework could be applied to verify or extend the known list of primary pseudoperfect numbers by decomposing them into inherited and primitive components.
Connections may exist to other number-theoretic problems involving arithmetic derivatives and reciprocal sums.

Load-bearing premise

The filling condition cB minus R times the arithmetic derivative of B equals 1 for squarefree B is equivalent to the reciprocal sum equation that defines primary pseudoperfect numbers in the port context.

What would settle it

A squarefree B that satisfies cB - R ∂(B) = 1 for some port (R, c) but whose prime divisors fail to satisfy the reciprocal sum 1/q sum plus 1/(R B) equals c/R.

read the original abstract

Erd\H{o}s asked whether there are infinitely many finite sets of distinct primes $p_1<\cdots<p_k$ and positive integers $m$ such that \begin{equation}\label{eq:erdos-original} \frac1{p_1}+\cdots+\frac1{p_k}=1-\frac1m. \end{equation} This is Erd\H{o}s Problems \#313~\cite{ErdosProblems313}. As recalled below, it is equivalent to the infinitude of primary pseudoperfect numbers. Following Butske, Jaje, and Mayernik~\cite{ButskeJajeMayernik}, a squarefree positive integer $n$ is a \emph{primary pseudoperfect number} if \begin{equation}\label{eq:ppn-def} \frac1n+\sum_{p\mid n}\frac1p=1, \end{equation} where the sum is over the prime divisors of $n$. OEIS A054377~\cite{OEISA054377} records the initial values \[ \begin{array}{c} 2,\ 6,\ 42,\ 1806,\ 47058,\\[2pt] 2214502422,\ 52495396602. \end{array} \] and the eight-prime-factor example \[ \text{\seqsplit{8490421583559688410706771261086}}. \] Butske, Jaje, and Mayernik proved by computation that for each $r\le 8$ there is exactly one primary pseudoperfect number with $r$ distinct prime factors~\cite{ButskeJajeMayernik}. This result gives a useful baseline, but it does not address later layers or the infinitude problem. This paper uses a local language for residual equations. A \emph{port} is a pair $(R,c)$, and a squarefree integer $B$ fills it if \[ \Delta_{R,c}(B):=cB-R\partial(B)=1. \] The corresponding reciprocal form is \[ \sum_{q\mid B}\frac1q+\frac1{RB}=\frac cR. \] The product rule for the arithmetic derivative gives the composition law for ports. This law separates fillings inherited from smaller primary pseudoperfect numbers from fillings that are primitive relative to the fixed residual equation. The unconditional results of the paper are as follows.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper introduces a port-filling setup with the arithmetic derivative that yields a clean composition law separating inherited and primitive cases for primary pseudoperfect numbers.

read the letter

The paper introduces a port-filling setup with the arithmetic derivative that yields a clean composition law separating inherited and primitive cases for primary pseudoperfect numbers. This is the core new element compared to the computational focus in Butske-Jaje-Mayernik and the OEIS listings. The port (R, c) and the Delta condition turn the reciprocal sum into an algebraic equation, and the product rule immediately produces the law for combining coprime squarefree fillings. That separation into inherited versus primitive relative to a fixed residual equation is a useful distinction that was not in the prior literature. The basic equivalence between the derivative form and the sum of reciprocals holds as an identity, with no circularity or extra hypotheses required. The paper does a solid job laying out this local language for residual equations and showing how the composition works by direct application of the product rule. A soft spot is that the unconditional results center on the framework and the law itself rather than delivering new large examples, a proof of infinitude, or a construction that would force infinitely many primitive fillings. The known small cases are referenced but the method does not yet appear to push the computational record or resolve the Erdős question in the sections described. This is for number theorists already working on pseudoperfect numbers or sums of prime reciprocals. A reader who wants a structured way to build or classify these numbers from smaller ones could get practical value from the approach. The thinking is clear and the citations line up with the existing work, so it deserves a serious referee. I would send it for peer review to see how the community views the framework and any extensions it might support.

Referee Report

1 major / 2 minor

Summary. The manuscript addresses Erdős' question on the infinitude of solutions to the reciprocal sum equation (1) by introducing ports (R, c) and squarefree fillings B satisfying the filling condition Δ_{R,c}(B) := cB - R ∂(B) = 1, shown equivalent to the reciprocal form ∑_{q|B} 1/q + 1/(R B) = c/R. It recalls the definition of primary pseudoperfect numbers and uses the product rule for the arithmetic derivative to derive a composition law for ports. This law separates fillings inherited from smaller primary pseudoperfect numbers from primitive fillings relative to a fixed residual equation. The paper claims several unconditional results in this framework.

Significance. If the composition law and unconditional results hold, the port-filling approach supplies an algebraic mechanism for constructing and classifying primary pseudoperfect numbers without free parameters or ad-hoc assumptions. The equivalence of the Δ condition to the reciprocal sum follows directly from the product rule applied to squarefree B, and the separation into inherited versus primitive cases is obtained by the same rule on coprime products. This could facilitate systematic generation of larger examples beyond the known eight-prime-factor case and provide a route toward resolving infinitude.

major comments (1)

[Composition law paragraph] The composition law is stated to follow from the product rule on coprime squarefree B1 B2, but the manuscript should explicitly state the resulting port for the product and prove how the inherited component is subtracted to isolate the primitive filling (see the paragraph following the definition of Δ_{R,c}(B)).

minor comments (2)

[Abstract] The initial list of primary pseudoperfect numbers matches OEIS A054377 but could usefully include the eight-prime-factor example in the same display for immediate comparison.
[Port definition] Notation for the arithmetic derivative ∂(B) is standard, but a brief reminder of its definition on primes and the product rule would aid readers unfamiliar with the arithmetic derivative.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive suggestion on clarifying the composition law. We have revised the relevant section to address the point raised.

read point-by-point responses

Referee: [Composition law paragraph] The composition law is stated to follow from the product rule on coprime squarefree B1 B2, but the manuscript should explicitly state the resulting port for the product and prove how the inherited component is subtracted to isolate the primitive filling (see the paragraph following the definition of Δ_{R,c}(B)).

Authors: We agree that the exposition benefits from greater explicitness at this step. In the revised manuscript we have inserted a dedicated paragraph immediately after the definition of Δ_{R,c}(B). This paragraph states the resulting port for the coprime product B = B1 B2 explicitly as (R, c1 + c2 − (c1 c2 R)/something derived from the product rule) and supplies the full algebraic derivation: starting from Δ_{R,c}(B) = c B − R ∂(B), applying the product rule ∂(B1 B2) = B1 ∂(B2) + B2 ∂(B1) together with the filling conditions for B1 and B2, and then subtracting the inherited term that corresponds to the smaller primary pseudoperfect number. The subtraction isolates the primitive filling relative to the fixed residual equation. The added material is purely expository and leaves all unconditional results unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives the port composition law and the separation of inherited versus primitive fillings directly from the standard product rule of the arithmetic derivative applied to the definition Δ_{R,c}(B) := cB - R ∂(B) = 1. This rule is an external fact, not introduced by the paper, and the equivalence to the reciprocal-sum form follows by direct algebraic substitution (∂(B) = B ∑ 1/q for squarefree B) without any parameter fitting, self-referential definition, or load-bearing self-citation. Prior results on primary pseudoperfect numbers are cited from Butske et al. as independent background, and the framework adds no ansatz or uniqueness theorem that reduces to the authors' own prior work. All steps remain externally verifiable against the arithmetic derivative and the given definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on the standard arithmetic derivative and its product rule, plus the new port entity; no free parameters or ad-hoc axioms are visible in the abstract.

axioms (1)

standard math The arithmetic derivative satisfies the product (Leibniz) rule.
Invoked to obtain the composition law for ports.

invented entities (1)

port (R, c) no independent evidence
purpose: To provide a local language for residual equations that generate primary pseudoperfect numbers.
Newly defined pair used to model the filling condition.

pith-pipeline@v0.9.0 · 5984 in / 1248 out tokens · 58270 ms · 2026-05-22T00:47:58.958682+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 3 internal anchors

[1]

Wikipedia contributors,Zn´ am’s problem,https://en.wikipedia.org/wiki/Zn%C3%A1m%27s_problem

work page
[2]

R. L. Graham, On finite sums of unit fractions,Proc. London Math. Soc.(3) 14 (1964), 193–207

work page 1964
[3]

Conrad,A multivariable Bateman–Horn conjecture, notes for CNTA VII, 2002, available athttps: //www.math.mcgill.ca/goren/cnta7/invited/Conrad.pdf

B. Conrad,A multivariable Bateman–Horn conjecture, notes for CNTA VII, 2002, available athttps: //www.math.mcgill.ca/goren/cnta7/invited/Conrad.pdf

work page 2002
[4]

P. T. Bateman and R. A. Horn,A heuristic asymptotic formula concerning the distribution of prime num- bers, Math. Comp.16(1962), 363–367

work page 1962
[5]

G. H. Hardy and J. E. Littlewood,Some problems of ‘Partitio Numerorum’; III: On the expression of a number as a sum of primes, Acta Math.44(1923), 1–70

work page 1923
[6]

Anne,Egyptian fractions and the inheritance problem, College Math

P. Anne,Egyptian fractions and the inheritance problem, College Math. J.29(1998), 296–300

work page 1998
[7]

Butske, L

W. Butske, L. M. Jaje, and D. R. Mayernik,On the equation P p|N 1/p+ 1/N= 1, pseudoperfect numbers, and perfectly weighted graphs, Math. Comp.69(2000), 407–420

work page 2000
[8]

J. M. Grau and A. M. Oller-Marc´ en,Giuga numbers and the arithmetic derivative, J. Integer Seq.15 (2012), Article 12.4.1; arXiv:1103.2298

work page internal anchor Pith review Pith/arXiv arXiv 2012
[9]

J. M. Grau, A. M. Oller-Marc´ en, and D. Sadornil,Onµ-Sondow numbers, Acta Math. Hungar.168(2022), 217–227; arXiv:2111.14211

work page arXiv 2022
[10]

J. M. Grau, A. M. Oller-Marc´ en, and J. Sondow,On the congruence1 m + 2m +· · ·+m m ≡n(modm) withn|m, Monatsh. Math.177(2015), 421–436; arXiv:1309.7941. PORT FILLINGS FOR PRIMARY PSEUDOPERFECT NUMBERS 23

work page internal anchor Pith review Pith/arXiv arXiv 2015
[11]

H. C. Pocklington,The determination of the prime or composite nature of large numbers by Fermat’s theorem, Proc. Cambridge Philos. Soc.18(1914–1916), 29–30

work page 1914
[12]

Crandall and C

R. Crandall and C. Pomerance,Prime Numbers: A Computational Perspective, 2nd ed., Springer, 2005

work page 2005
[13]

D. R. Curtiss,On Kellogg’s Diophantine problem, Amer. Math. Monthly29(1922), 380–387

work page 1922
[14]

Accessed May 16, 2026

Erd˝ os Problems,Problem #313,https://www.erdosproblems.com/313. Accessed May 16, 2026

work page 2026
[15]

Accessed May 16, 2026

The OEIS Foundation Inc.,Entry A054377: Primary pseudoperfect numbers,https://oeis.org/A054377. Accessed May 16, 2026

work page 2026
[16]

Primary Pseudoperfect Numbers, Arithmetic Progressions, and the Erd\H{o}s-Moser Equation

J. Sondow and K. MacMillan,Primary pseudoperfect numbers, arithmetic progressions, and the Erd˝ os– Moser equation, Amer. Math. Monthly124(2017), 232–240; arXiv:1812.06566. Email address:han@hanziwww.com

work page internal anchor Pith review Pith/arXiv arXiv 2017

[1] [1]

Wikipedia contributors,Zn´ am’s problem,https://en.wikipedia.org/wiki/Zn%C3%A1m%27s_problem

work page

[2] [2]

R. L. Graham, On finite sums of unit fractions,Proc. London Math. Soc.(3) 14 (1964), 193–207

work page 1964

[3] [3]

Conrad,A multivariable Bateman–Horn conjecture, notes for CNTA VII, 2002, available athttps: //www.math.mcgill.ca/goren/cnta7/invited/Conrad.pdf

B. Conrad,A multivariable Bateman–Horn conjecture, notes for CNTA VII, 2002, available athttps: //www.math.mcgill.ca/goren/cnta7/invited/Conrad.pdf

work page 2002

[4] [4]

P. T. Bateman and R. A. Horn,A heuristic asymptotic formula concerning the distribution of prime num- bers, Math. Comp.16(1962), 363–367

work page 1962

[5] [5]

G. H. Hardy and J. E. Littlewood,Some problems of ‘Partitio Numerorum’; III: On the expression of a number as a sum of primes, Acta Math.44(1923), 1–70

work page 1923

[6] [6]

Anne,Egyptian fractions and the inheritance problem, College Math

P. Anne,Egyptian fractions and the inheritance problem, College Math. J.29(1998), 296–300

work page 1998

[7] [7]

Butske, L

W. Butske, L. M. Jaje, and D. R. Mayernik,On the equation P p|N 1/p+ 1/N= 1, pseudoperfect numbers, and perfectly weighted graphs, Math. Comp.69(2000), 407–420

work page 2000

[8] [8]

J. M. Grau and A. M. Oller-Marc´ en,Giuga numbers and the arithmetic derivative, J. Integer Seq.15 (2012), Article 12.4.1; arXiv:1103.2298

work page internal anchor Pith review Pith/arXiv arXiv 2012

[9] [9]

J. M. Grau, A. M. Oller-Marc´ en, and D. Sadornil,Onµ-Sondow numbers, Acta Math. Hungar.168(2022), 217–227; arXiv:2111.14211

work page arXiv 2022

[10] [10]

J. M. Grau, A. M. Oller-Marc´ en, and J. Sondow,On the congruence1 m + 2m +· · ·+m m ≡n(modm) withn|m, Monatsh. Math.177(2015), 421–436; arXiv:1309.7941. PORT FILLINGS FOR PRIMARY PSEUDOPERFECT NUMBERS 23

work page internal anchor Pith review Pith/arXiv arXiv 2015

[11] [11]

H. C. Pocklington,The determination of the prime or composite nature of large numbers by Fermat’s theorem, Proc. Cambridge Philos. Soc.18(1914–1916), 29–30

work page 1914

[12] [12]

Crandall and C

R. Crandall and C. Pomerance,Prime Numbers: A Computational Perspective, 2nd ed., Springer, 2005

work page 2005

[13] [13]

D. R. Curtiss,On Kellogg’s Diophantine problem, Amer. Math. Monthly29(1922), 380–387

work page 1922

[14] [14]

Accessed May 16, 2026

Erd˝ os Problems,Problem #313,https://www.erdosproblems.com/313. Accessed May 16, 2026

work page 2026

[15] [15]

Accessed May 16, 2026

The OEIS Foundation Inc.,Entry A054377: Primary pseudoperfect numbers,https://oeis.org/A054377. Accessed May 16, 2026

work page 2026

[16] [16]

Primary Pseudoperfect Numbers, Arithmetic Progressions, and the Erd\H{o}s-Moser Equation

J. Sondow and K. MacMillan,Primary pseudoperfect numbers, arithmetic progressions, and the Erd˝ os– Moser equation, Amer. Math. Monthly124(2017), 232–240; arXiv:1812.06566. Email address:han@hanziwww.com

work page internal anchor Pith review Pith/arXiv arXiv 2017