Pairs (x,y); d(x) = d(y), σ(x) = σ(y), φ(x) = φ(y)

 
 

638   has 8 divisors:     1, 2, 11, 22, 29, 58, 319, 638
Sum of divisors:     1080

568   has 8 divisors:     1, 2, 4, 8, 71, 142, 284, 568
Sum of divisors:     1080
 
 
d \, (568) \; = \; d \, (638) \; = \; 8

\sigma \, (568) \; = \; \sigma \, (638) \; = \; 1080

\phi \, (568) \; = \; \phi \, (638) \; = \; 280

 

Find other pair of integers   x,   y,   y > x   such that

d \, (x) \; = \; d \, (y)
\sigma \, (x) \; = \; \sigma \, (y)
\phi \, (x) \; = \; \phi \, (y)

 
 
All equations are satisfied by the numbers
x \; = \; 3^{n} \; \times \; 568    and    y \; = \; 3^{n} \; \times \; 638,   where   n = 0, 1, 2, 3, …

 
Also by these multiples:

x = 2840 = 5 × 568,         y = 3190 = 5 × 638
x = 3976 = 7 × 568,          y = 4466 = 7 × 638
x = 7384 = 13 × 568,        y = 8294 = 13 × 638
x = 8520 = 15 × 568,        y = 9570 = 15 × 638
x = 10792 = 19 × 568,      y = 12122 = 19 × 638
x = 11928 = 21 × 568,      y = 13398 = 21 × 638
x = 13064 = 23 × 568,      y = 14674 = 23 × 638
…………………………………………………
…………………………………………………

 
(x, y) = (1824, 1836)   and some of its multiples

x = 9120 = 5 × 1824,         y = 9180 = 5 × 1836

 
(x, y)   =   (1704, 1914)

x = 5112 = 3 × 1704,         y = 5742 = 3 × 1914

 
(x, y)   =   (4185 , 4389)

x = 8370 = 2 × 4185,         y = 8778 = 2 × 4389

 
(x, y)   =   (3051, 3219)

x = 6102 = 2 × 3051,         y = 6438 = 2 × 3219

 
(x, y)   =   (4960, 5236)

x = 14880 = 3 × 4960,         y = 15708 = 3 × 5236

 
(x, y)   =  (6368, 6764)

x = 19104 = 3 × 6368,         y = 20292 = 3 × 6764

 
(x, y)   =   (7749, 8151)

x = 15498 = 2 × 7749,         y = 16302 = 2 × 8151

 
(x, y)   =   (9184, 9724)

x = 27552 = 3 × 9184,         y = 29172 = 3 × 9724

 
 
 
 
 
 
 
 
 
 
 
 
 
 

 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

3 Responses to Pairs (x,y); d(x) = d(y), σ(x) = σ(y), φ(x) = φ(y)

  1. pipo says:

    Here are some more:
    Format : pair, nr of divisors, sum of divisors, totient)
    ( 568 , 638 ) 8 1080 280
    ( 1704 , 1914 ) 16 4320 560
    ( 3051 , 3219 ) 8 4560 2016
    ( 1824 , 1836 ) 24 5040 576
    ( 2840 , 3190 ) 16 6480 1120
    ( 4185 , 4389 ) 16 7680 2160
    ( 3976 , 4466 ) 16 8640 1680
    ( 4960 , 5236 ) 24 12096 1920
    ( 6368 , 6764 ) 12 12600 3168
    ( 7749 , 8151 ) 16 13440 4320
    ( 6102 , 6438 ) 16 13680 2016
    ( 5112 , 5742 ) 24 14040 1680
    ( 7384 , 8294 ) 16 15120 3360
    ( 9184 , 9724 ) 24 21168 3840
    ( 8370 , 8778 ) 32 23040 2160
    ( 8520 , 9570 ) 32 25920 2240
    ( 9120 , 9180 ) 48 30240 2304

    pipo

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