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[David W. Wilson]

N. J. A. Sloane wrote:

> 3. There are some excellent new sequences (besides the usual bunch of clunkers(*))
> at the end of the Big Table, many of which need extending (e.g. 37171, 37181, 37195...,
> 37207, 37211 etc.).  Look for the keywords "more" and "nice".

A037171 is an interesting one.  It is basically the numbers for which phi(n) = pi(n).
I am sure the sequence is finite, and would be surprised if the published sequence is
not complete.

I more or less attacked it this way:

I believe that the value of phi(n)/n drops at n = p# for prime p.  Therefore, using the
approximation pi(n) = n/log n, I took the values of pi(n)/n and phi(n)/n at the primorials,
getting the following table:

       p    pi(p#)/p#   phi(p#)/p#
       2    1.442695    0.500000
       3    0.455120    0.333333
       5    0.207112    0.266667
       7    0.128475    0.228571
      11    0.083406    0.207792
      13    0.064979    0.191808
      17    0.050422    0.180525
      19    0.042453    0.171024
      23    0.035437    0.163588
      29    0.029697    0.157947
      31    0.026473    0.152852
      37    0.023078    0.148721
      41    0.020714    0.145094
      43    0.018991    0.141719
      47    0.017315    0.138704
      53    0.015742    0.136087
      59    0.014426    0.133780
      61    0.013514    0.131587
      67    0.012517    0.129623
      71    0.011730    0.127798
      73    0.011099    0.126047
      79    0.010403    0.124451
      83    0.009839    0.122952
      89    0.009283    0.121571
      97    0.008744    0.120317

Thus is looks as if pi(n) is growing much more slowly than phi(n) relative n, as one
would expect.  I highly suspect that pi(n) < phi(n) (the real pi(n)) for sufficient n
could be proved, though I don't have the analysis skills to prove it.  This would
imply that A037171 is finite.  Actually, I wouldn't be surprised if A037171 is complete
as it stands.