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erdős #315

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

Worked, still open.

number theory · solved · 0 attempts

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A000058 — Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(0) = 2.2,3,7,43,1807,3263443,10650056950807,113423713055421844361000443,12864938683278671740537145998360961546653259485195807 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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Sylvester's sequence · reference

Create a formalisation here · link

F (k)

be the number of solutions to

1 = \frac{1}{n _{1}} + \dots + \frac{1}{n _{k}},

1 \leq n_{1} < \dots < n_{k}

are distinct integers. Find good estimates for

F (k)

.A076393 #243Let

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

be a sequence of integers such that

n \to \infty lim \frac{a _{n}}{a _{n - 1}^{2}} = 1

\sum \frac{1}{a _{n}} \in Q

. Then, for all sufficiently large

n \geq 1

a_{n} = a_{n - 1}^{2} - a_{n - 1} + 1.

status

solved

finding.noted · reviewer:will-blair · 1 day

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