Erdős Problem #1054
Let f(n) be the minimal integer m such that n is the sum of the k smallest divisors of m for some k≥ 1. Is it true that f(n)=o(n)?
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Open problems about divisors
Problems on the divisors of integers: their distribution, consecutive divisors, divisor sums and highly composite numbers. Most come from Erdős's lists and can be explored numerically before a proof is attempted.
Erdős Problem #18
Conjecture 1. Are there infinitely many practical numbers m such that h(m) < (log log m)^O(1)? More precisely: does there exist a constant C > 0 such that for infinitely many practical numbers m, we have h(m) < (log log m)^C?
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Erdős Problem #450
How large must y=y(ε,n) be such that the number of integers in (x,x+y) with a divisor in (n,2n) is at most ε y? The bound is required for every x and every window length at least y, and y(ε,n) is the least such threshold (or ∞ if there is none).
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Erdős Problem #470
Are there any odd weird numbers?
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Erdős Problem #859
The density of the divisor sum set is asymptotically equivalent to c_1 / log(t)^c_2.
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Erdős Problem #885
Is it true that, for every k ≥ 1, there exist integers N_1 < … < N_k such that |∩_i D(N_i)| ≥ k?
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Erdős Problem #886
Let ε>0. Is it true that, for all large n, the number of divisors of n in (n^1/2,n^1/2+n^1/2-ε) is O_ε(1)? Erdős attributes this conjecture to Ruzsa.
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Erdős Problem #887
Is there an absolute constant K such that, for every C > 0, if n is sufficiently large then n has at most K divisors in (n^1/2, n^1/2 + C n^1/4).
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Erdős Problem #893
Does the limit lim_n→∞ f(2n)/f(n) tend to infinity? (Other finite limits have been ruled out by [KoLu25], see below)
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Erdős Problem #945
Is it true that F(x) ≤ (log x)^O(1)?
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Erdős Problem #975
For an irreducible polynomial f ∈ ℤ[x] with f(n) ≥ 1 for sufficiently large n, does there exists a constant c = c(f) > 0 such that Σ_n ≤ x τ(f(n)) ≈ c · x log x? Note that it is unclear whether the polynomial should have integer coefficients or merely be integer-valued. We assume the former.
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