Infinite families and sporadic solutions of the equation $φ(n)=c+α\sum_{p\mid n}φ(p)$

Abstract

Ranjan and Singh proved that for every composite $n$ which is not twice an odd prime, $φ(n)$ is at least one plus the sum of $φ$ over the prime divisors of $n$, with equality exactly for $n=4,12,30$. \new{We study the weighted equation $φ(n)=c+α\sum_{p\mid n}φ(p)$. We determine the parameters $(c,α)$ admitting infinite families of solutions, prove that all remaining solutions are finite in number and arise from an explicit linear mechanism, and compute them exactly in a range of parameters; the finiteness of $\{4,12,30\}$ becomes a special case. We also settle the analogous equations for the divisor functions $τ$ and $σ$ and for the Jordan totients, and, from known results on the sum of the prime divisors, derive the full asymptotic expansion of the average deficiency.} \K{In the range $|c|\le10$, $1\leα\le6$ the composite solution set is determined completely\V{, and the number and the size of the sporadic solutions are shown to be bounded by polynomials in $|c|+α$}.} \B{Finally, replacing the prime values $φ(p)$ by their iterates $φ^{(k)}(p)$, we determine all solutions of $φ(n)=1+\sum_{p\mid n}φ^{(k)}(p)$: the solution sets are $\{1,4,12,30\}$, $\{1,3,4,10\}$ and, for every $k\ge3$, $\{1,3,4\}$; for the weighted iterated equation we exhibit an infinite family governed by the safe primes.