Let $F (k)$ be the number of solutions to $1 = \frac{1}{n _{1}} + \dots + \frac{1}{n _{k}},$ where $1 \leq n_{1} < \dots < n_{k}$ are distinct integers. Find good estimates for $F (k)$ .

Open problem — our best result is machine-sealed: verified reduction, reproduced by an independent verifier. The conjecture itself is unsettled.

number theory · open · 1 attempt

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

verified reduction

needs verification

Transfer from #306: exact F(k) (number of representations of 1 by k distinct unit fractions) for k<=7 = [F(1)=1,F(2)=0,F(3)=1,F(4)=6,F(5)=72,F(6)=2320,F(7)=245765]; split-tree lower-bound table.

#306 split-family viewpoint: collision-free 1/n=1/(n+1)+1/(n(n+1)) grows term count by one. SCOPE: exact exhaustive counts k<=7; bounded split-tree lower bounds for larger k; does NOT improve known asymptotics.

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

[ F(k)=#\\{(n_1,\dots,n_k): 1\le n_1<\cdots<n_k,;;1=\sum_{i=1}^k \frac1{n_i}\\}. ]

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

candidate solution ↗

oeis

A006585 — Egyptian fractions: number of solutions to 1 = 1/x_1 + ... + 1/x_n in positive integers x_1 < ... < x_n.1,0,1,6,72,2320,245765,151182379 A076393 — Decimal expansion of Vardi constant arising in the Sylvester sequence.1,2,6,4,0,8,4,7,3,5,3,0,5,3,0,1,1,1,3,0,7,9,5,9,9,5,8,4,1,6,4,6,6,9,4,9,1,1,1,4,5,6,0,1,7,9,2,0,9,0,6,5,5,3,3,1,5,3,4,5,

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related: #306

#315Let

u_{1} = 1

and

u_{n + 1} = u_{n} (u_{n} + 1)

, so that

\sum_{k \geq 1} \frac{1}{u _{k} + 1}

and

u_{k} = ⌊ c_{0}^{2^{k}} + 1 ⌋

for

k \geq 1

, where

c_{0} = lim u_{n}^{1/ 2^{n}} = 1.264085 \dots .

Let

a_{1} < a_{2} < \dots

be any other sequence with

\sum \frac{1}{a _{k}} = 1

. Is it true that

lim inf a_{n}^{1/ 2^{n}} < c_{0} = 1.264085 \dots ?

A076393

status

open

evidence

oeis

links

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