<none>
David W. Wilson
wilson at cabletron.com
Mon Jun 15 16:58:29 CEST 1998
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.
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