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Source. Wouter van Doorn and GPT-6 Astra Pro (the author line as printed), Practical numbers and Egyptian fractions, Corollary 3.4 with its proof, physical p. 5 of the seven-page PDF held by van Doorn (2026). Read on the page image. Uses the Lemma 3.1 reconstruction and the Lemma 3.3 reconstruction; consumed by the Proposition 4.1 reconstruction.

Standing. Author-recorded reconstruction of a claimed result (see the Lemma 3.1 page for the note's standing); not an independent review; changes no status and assigns no tier.

Statement

Under the hypotheses of Lemma 3.3 (so $k$ is large, $p_*$ is an odd prime, and $V$ is odd, squarefree, with $\omega(V)=k$ and all prime factors at most $2Q(k)$), suppose that $E\ge4$ and that $n=2^EV$ is practical. Then the modulus $A$ supplied by Lemma 3.3 satisfies

$$An\text{ is practical},\qquad h(An)\le h(n)+4 .$$

Proof

Let $c$ be a residue modulo $A$. Lemma 3.3 gives divisors $z_0,z_1,z_2,z_3$ of $V$ with $c\equiv z_0+2z_1+4z_2+8z_3\pmod A$. Consider the four integers $2^\ell z_\ell$, $\ell=0,1,2,3$. Each divides $2^EV=n$, since $\ell\le3<E$ and $z_\ell\mid V$. Each is coprime to $A$, since $A$ is odd and coprime to $V$; in particular none is divisible by $A$. They are distinct, since $V$ is odd, so the $z_\ell$ are odd and the $2$-adic valuation of $2^\ell z_\ell$ is exactly $\ell$. Their sum is at most $(1+2+4+8)V=15V<16V\le2^EV=n$. Thus every residue modulo $A$ is represented by a sum of at most $4$ distinct divisors of $n$ with total at most $n$ and no summand divisible by $A$, and $A\ge2$. Lemma 3.1 with $L=4$ gives the conclusion.