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erdős #1064

← #1063 · #1065 → (packet.json; erdosproblems.com)

Prove that $ϕ (n) > ϕ (n - ϕ (n))$ for almost all $n$ , but that $ϕ (n) < ϕ (n - ϕ (n))$ for infinitely many $n$ , where $ϕ$ is Euler's totient function.

Worked, still open.

number theory · solved · formalized (Lean) · 0 attempts

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formal

AMS 11 · open (literature)

theorem erdos_1106.parts.i :
    answer(sorry) ↔ Tendsto (fun n => #(∏ i ∈ Icc 1 n, p i).primeFactors) atTop atTop

formal-conjectures/1064.lean ↗

oeis

A051487 — Numbers k such that phi(k) = phi(k - phi(k)).2,6,12,24,48,96,150,192,300,384,600,726,750,768,1200,1452,1500,1536,2310,2400,2904,3000,3072,3174,3750,4620,4800,5046,58 A051488 — Numbers k such that phi(k) < phi(k - phi(k)).30,60,66,120,132,138,174,210,240,246,264,276,318,330,348,420,480,492,498,510,528,534,552,630,636,660,678,690,696,786,840

status

solved

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

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