Consecutivos idénticos valores de $\phi(n)$

Question

Hay un número infinito de $n$$\phi(n) = \phi(n+1)$? Algunos ejemplos son los $n \in \{3, 15, 104, 65535\}$.

Answer 1

1 =  1    2 = 2 phi 1 =  1 
3 = 3   4 = 2^2 phi 2 = 2
15 = 3 * 5   16 = 2^4 phi 8 = 2^3
104 = 2^3 * 13   105 = 3 * 5 * 7 phi 48 = 2^4 * 3
164 = 2^2 * 41   165 = 3 * 5 * 11 phi 80 = 2^4 * 5
194 = 2 * 97   195 = 3 * 5 * 13 phi 96 = 2^5 * 3
255 = 3 * 5 * 17   256 = 2^8 phi 128 = 2^7
495 = 3^2 * 5 * 11   496 = 2^4 * 31 phi 240 = 2^4 * 3 * 5
584 = 2^3 * 73   585 = 3^2 * 5 * 13 phi 288 = 2^5 * 3^2
975 = 3 * 5^2 * 13   976 = 2^4 * 61 phi 480 = 2^5 * 3 * 5
2204 = 2^2 * 19 * 29   2205 = 3^2 * 5 * 7^2 phi 1008 = 2^4 * 3^2 * 7
2625 = 3 * 5^3 * 7   2626 = 2 * 13 * 101 phi 1200 = 2^4 * 3 * 5^2
2834 = 2 * 13 * 109   2835 = 3^4 * 5 * 7 phi 1296 = 2^4 * 3^4
3255 = 3 * 5 * 7 * 31   3256 = 2^3 * 11 * 37 phi 1440 = 2^5 * 3^2 * 5
3705 = 3 * 5 * 13 * 19   3706 = 2 * 17 * 109 phi 1728 = 2^6 * 3^3
5186 = 2 * 2593   5187 = 3 * 7 * 13 * 19 phi 2592 = 2^5 * 3^4
5187 = 3 * 7 * 13 * 19   5188 = 2^2 * 1297 phi 2592 = 2^5 * 3^4
10604 = 2^2 * 11 * 241   10605 = 3 * 5 * 7 * 101 phi 4800 = 2^6 * 3 * 5^2
11715 = 3 * 5 * 11 * 71   11716 = 2^2 * 29 * 101 phi 5600 = 2^5 * 5^2 * 7
13365 = 3^5 * 5 * 11   13366 = 2 * 41 * 163 phi 6480 = 2^4 * 3^4 * 5
18315 = 3^2 * 5 * 11 * 37   18316 = 2^2 * 19 * 241 phi 8640 = 2^6 * 3^3 * 5
22935 = 3 * 5 * 11 * 139   22936 = 2^3 * 47 * 61 phi 11040 = 2^5 * 3 * 5 * 23
25545 = 3 * 5 * 13 * 131   25546 = 2 * 53 * 241 phi 12480 = 2^6 * 3 * 5 * 13
32864 = 2^5 * 13 * 79   32865 = 3 * 5 * 7 * 313 phi 14976 = 2^7 * 3^2 * 13
38804 = 2^2 * 89 * 109   38805 = 3 * 5 * 13 * 199 phi 19008 = 2^6 * 3^3 * 11
39524 = 2^2 * 41 * 241   39525 = 3 * 5^2 * 17 * 31 phi 19200 = 2^8 * 3 * 5^2
46215 = 3^2 * 5 * 13 * 79   46216 = 2^3 * 53 * 109 phi 22464 = 2^6 * 3^3 * 13
48704 = 2^6 * 761   48705 = 3 * 5 * 17 * 191 phi 24320 = 2^8 * 5 * 19
49215 = 3 * 5 * 17 * 193   49216 = 2^6 * 769 phi 24576 = 2^13 * 3
49335 = 3 * 5 * 11 * 13 * 23   49336 = 2^3 * 7 * 881 phi 21120 = 2^7 * 3 * 5 * 11
56864 = 2^5 * 1777   56865 = 3 * 5 * 17 * 223 phi 28416 = 2^8 * 3 * 37
57584 = 2^4 * 59 * 61   57585 = 3 * 5 * 11 * 349 phi 27840 = 2^6 * 3 * 5 * 29
57645 = 3^3 * 5 * 7 * 61   57646 = 2 * 19 * 37 * 41 phi 25920 = 2^6 * 3^4 * 5
64004 = 2^2 * 16001   64005 = 3 * 5 * 17 * 251 phi 32000 = 2^8 * 5^3
65535 = 3 * 5 * 17 * 257   65536 = 2^16 phi 32768 = 2^15
73124 = 2^2 * 101 * 181   73125 = 3^2 * 5^4 * 13 phi 36000 = 2^5 * 3^2 * 5^3
105524 = 2^2 * 23 * 31 * 37   105525 = 3^2 * 5^2 * 7 * 67 phi 47520 = 2^5 * 3^3 * 5 * 11
107864 = 2^3 * 97 * 139   107865 = 3^3 * 5 * 17 * 47 phi 52992 = 2^8 * 3^2 * 23
123824 = 2^4 * 71 * 109   123825 = 3 * 5^2 * 13 * 127 phi 60480 = 2^6 * 3^3 * 5 * 7
131144 = 2^3 * 13^2 * 97   131145 = 3 * 5 * 7 * 1249 phi 59904 = 2^9 * 3^2 * 13
164175 = 3 * 5^2 * 11 * 199   164176 = 2^4 * 31 * 331 phi 79200 = 2^5 * 3^2 * 5^2 * 11
184635 = 3^2 * 5 * 11 * 373   184636 = 2^2 * 31 * 1489 phi 89280 = 2^6 * 3^2 * 5 * 31
198315 = 3^3 * 5 * 13 * 113   198316 = 2^2 * 43 * 1153 phi 96768 = 2^9 * 3^3 * 7
214334 = 2 * 31 * 3457   214335 = 3^2 * 5 * 11 * 433 phi 103680 = 2^8 * 3^4 * 5
215775 = 3^2 * 5^2 * 7 * 137   215776 = 2^5 * 11 * 613 phi 97920 = 2^7 * 3^2 * 5 * 17
256274 = 2 * 97 * 1321   256275 = 3^2 * 5^2 * 17 * 67 phi 126720 = 2^8 * 3^2 * 5 * 11
286995 = 3 * 5 * 19^2 * 53   286996 = 2^2 * 157 * 457 phi 142272 = 2^6 * 3^2 * 13 * 19
307395 = 3^5 * 5 * 11 * 23   307396 = 2^2 * 31 * 37 * 67 phi 142560 = 2^5 * 3^4 * 5 * 11
319275 = 3^3 * 5^2 * 11 * 43   319276 = 2^2 * 19 * 4201 phi 151200 = 2^5 * 3^3 * 5^2 * 7
347324 = 2^2 * 31 * 2801   347325 = 3 * 5^2 * 11 * 421 phi 168000 = 2^6 * 3 * 5^3 * 7
388245 = 3 * 5 * 11 * 13 * 181   388246 = 2 * 17 * 19 * 601 phi 172800 = 2^8 * 3^3 * 5^2
397485 = 3^2 * 5 * 11^2 * 73   397486 = 2 * 23 * 8641 phi 190080 = 2^7 * 3^3 * 5 * 11
407924 = 2^2 * 11 * 73 * 127   407925 = 3^2 * 5^2 * 7^2 * 37 phi 181440 = 2^6 * 3^4 * 5 * 7
415275 = 3 * 5^2 * 7^2 * 113   415276 = 2^2 * 17 * 31 * 197 phi 188160 = 2^8 * 3 * 5 * 7^2
454124 = 2^2 * 11 * 10321   454125 = 3 * 5^3 * 7 * 173 phi 206400 = 2^6 * 3 * 5^2 * 43
491535 = 3^3 * 5 * 11 * 331   491536 = 2^4 * 31 * 991 phi 237600 = 2^5 * 3^3 * 5^2 * 11
524432 = 2^4 * 73 * 449   524433 = 3 * 7 * 13 * 17 * 113 phi 258048 = 2^12 * 3^2 * 7
525986 = 2 * 181 * 1453   525987 = 3^3 * 7 * 11^2 * 23 phi 261360 = 2^4 * 3^3 * 5 * 11^2
546272 = 2^5 * 43 * 397   546273 = 3^2 * 7 * 13 * 23 * 29 phi 266112 = 2^7 * 3^3 * 7 * 11
568815 = 3 * 5 * 13 * 2917   568816 = 2^4 * 73 * 487 phi 279936 = 2^7 * 3^7
589407 = 3 * 7 * 13 * 17 * 127   589408 = 2^5 * 113 * 163 phi 290304 = 2^9 * 3^4 * 7
679496 = 2^3 * 157 * 541   679497 = 3 * 7 * 13 * 19 * 131 phi 336960 = 2^6 * 3^4 * 5 * 13
686985 = 3 * 5 * 13^2 * 271   686986 = 2 * 53 * 6481 phi 336960 = 2^6 * 3^4 * 5 * 13
840255 = 3 * 5 * 13 * 31 * 139   840256 = 2^6 * 19 * 691 phi 397440 = 2^7 * 3^3 * 5 * 23
914175 = 3^2 * 5^2 * 17 * 239   914176 = 2^8 * 3571 phi 456960 = 2^8 * 3 * 5 * 7 * 17
936494 = 2 * 13 * 181 * 199   936495 = 3^3 * 5 * 7 * 991 phi 427680 = 2^5 * 3^5 * 5 * 11
952575 = 3 * 5^2 * 13 * 977   952576 = 2^8 * 61^2 phi 468480 = 2^9 * 3 * 5 * 61
983775 = 3 * 5^2 * 13 * 1009   983776 = 2^5 * 71 * 433 phi 483840 = 2^9 * 3^3 * 5 * 7
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