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Guy $(2004, p. 150)$ discussed solutions to $\phi(\sigma(n))=n $ ,In my below question I asked if he discussed also the solution of the below equation ,then I w'd like to seek for the pair of integers $( m, n)$ for which : $$\displaystyle \phi(\sigma(n))=m² $$ ?.

Note: $\displaystyle\sigma(n)=\sum_{d|n}d$ is sum divisor function ,and $\phi$ is Euler totiont function

Thank you for any help

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  • $\begingroup$ Usually $\sigma(n)$ denotes the sum divisor function of $n$, not "a power of sum divisor function". $\endgroup$ – Xam Dec 28 '16 at 0:09
  • $\begingroup$ yes, sorry for that and thanks for ur attention , i edited it now it's fixed $\endgroup$ – zeraoulia rafik Dec 28 '16 at 0:10
  • $\begingroup$ For what it's worth, numbers $k$ such that $\phi(k)$ is a square are tabulated at oeis.org/A039770 and thereare some links to the literature. $\endgroup$ – Gerry Myerson Dec 28 '16 at 1:26
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fairly common

=================================================

1   1
6   4
7   4
11   4
21   16
24   16
27   16
31   16
32   36
33   16
35   16
38   16
47   16
49   36
59   16
68   36
73   36
74   36
82   36
85   36
93   64
105   64
107   36
113   36
114   64
127   64
133   64
134   64
135   64
141   64
155   64
158   64
161   64
177   64
180   144
191   64
198   144
203   64
204   144
209   64
219   144
220   144
222   144
224   144
230   144
234   144
238   144
239   64
243   144
246   144
250   144
255   144
259   144
284   144
286   144
290   144
296   144
302   144
321   144
328   144
334   144
339   144
355   144
356   144
358   144
362   144
365   144
369   144
371   144
378   256
379   144
381   256
384   256
391   144
399   256
402   256
407   144
415   144
431   144
434   256
443   144
445   144
451   144
465   256
467   144
469   256
474   256
483   256
490   324
493   144
503   144
530   324
536   256
542   256
544   324
553   256
569   144
573   256
589   256
609   256
621   256
627   256
630   576
635   256
660   576
665   256
672   576
684   576
690   576
707   256
713   256
714   576
717   256
737   256
750   576
770   576
772   576
777   576
792   576
819   576
828   576
833   324
846   576
852   576
858   576
864   576
869   256
870   576
872   400
882   1296
888   576
893   256
899   256
901   324
906   576
910   576
918   576
920   576
936   576
940   576
952   576
971   324
977   324
984   576
992   576
994   576
998   400
999   576
1000   576

==============================================

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