✦ 5 frontiers

read-only vieweressays

erdős #468

← #467 · #469 → (packet.json; erdosproblems.com)

For any $n$ let $D_{n}$ be the set of sums of the shape $d_{1}, d_{1} + d_{2}, d_{1} + d_{2} + d_{3}, \dots$ where $1 < d_{1} < d_{2} < \dots$ are the divisors of $n$ . What is the size of $D_{n} \ \cup_{m < n} D_{m}$ ?If $f (N)$ is the minimal $n$ such that $N \in D_{n}$ then is it true that $f (N) = o (N)$ ? Perhaps just for almost all $N$ ?

Worked, still open.

number theory · open · 0 attempts

use this record

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

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

Let $ 1<d_1<d_2<\cdots<d_{k_n} $ be the divisors of $n$ larger than $1$, and $ D_n={d_1,\ d_1+d_2,\ \ldots,\ d_1+\cdots+d_{k_n}}. $

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

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

candidate solution ↗

oeis

A167485 — Smallest positive integer m such that n can be expressed as the sum of an initial subsequence of the divisors of m, or 0 if no such m exists.1,1,0,2,3,0,5,4,7,15,12,21,6,9,13,8,12,30,10,42,19,18,20,57,14,36,46,30,12,102,29,16,21,42,62,84,22,36,37,18,27,63,20,50 A387502 — Number of debut sums of initial subsequences of the divisors > 1 of n.0,1,1,1,1,1,1,1,1,1,0,3,1,1,1,1,0,2,1,2,2,1,0,1,0,0,1,1,1,3,0,1,1,1,0,4,1,0,0,2,0,3,1,1,1,1,0,3,1,2,0,2,0,2,0,1,2,0,0,3,A387503 — Total number of distinct sums of initial subsequences of the divisors > 1 of positive integers up to n.0,1,2,3,4,5,6,7,8,9,9,12,13,14,15,16,16,18,19,21,23,24,24,25,25,25,26,27,28,31,31,32,33,34,34,38,39,39,39,41,41,44,45,46

links

Create a formalisation here · link

f (n)

be the minimal integer

m

n

is the sum of the

k

smallest divisors of

m

k \geq 1

.Is it true that

f (n) = o (n)

? Or is this true only for almost all

n

lim sup f (n) / n = \infty

status

open

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

renders the record as of vev_d199cb2e · 1,338 events · hub