Infinite families and sporadic solutions of the equation φ(n)=c+α∑p| nφ(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∑p| nφ(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|≤10, 1≤α≤6 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+∑p| nφ(k)(p): the solution sets are {1,4,12,30}, {1,3,4,10} and, for every k≥3, {1,3,4}; for the weighted iterated equation we exhibit an infinite family governed by the safe primes.
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BibTeX
@misc{belbachir2026infinite,
title = {{Infinite families and sporadic solutions of the equation $φ(n)=c+α\sum_p\mid nφ(p)$}},
author = {Hacène Belbachir and Ravi Ranjan and Shubh N. Singh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36260},
eprint = {2610.36260},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36260},
note = {CC0},
}
RIS
TY - UNPB
TI - Infinite families and sporadic solutions of the equation φ(n)=c+α∑_p| nφ(p)
AU - Hacène Belbachir
AU - Ravi Ranjan
AU - Shubh N. Singh
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36260
UR - https://arxiv.org/abs/2610.36260
AB - 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∑_p| nφ(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|≤10, 1≤α≤6 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+∑_p| nφ^(k)(p): the solution sets are {1,4,12,30}, {1,3,4,10} and, for every k≥3, {1,3,4}; for the weighted iterated equation we exhibit an infinite family governed by the safe primes.
ER -