Erdős Problem #824
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
`h x` is the number of pairs of integers \(1 \leq a < b < x\) such that \((a, b) = 1\) and \(\sigma(a) = \sigma(b)\), where \(\sigma\) is the sum of divisors function. We take \(x\) to be a natural number. For real \(x\) the count equals `h ⌈x⌉`, so the asymptotic statements below do not change.
`hSquarefree x` is the number of pairs of integers \(1 \leq a < b < x\) such that \(a\) and \(b\) are squarefree and \(\sigma(a) = \sigma(b)\).
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Verified \(h(10^6)=6{,}437{,}809\), an explicit 46-member coprime sigma-fiber, and the sharper named-theorem bound \(h(x)>x^\alpha\) for every \(\alpha<1.4313775065\ldots\); exponent \(2\) still requi
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