[seqfan] Farey series of order 0

hv at crypt.org hv at crypt.org
Wed Dec 1 17:56:37 CET 2021


Since A005728 was added 30 years ago, it has given a(0) = 1 with a mostly
unchanged title "Number of fractions in Farey series of order n".

Can someone explain how a(0) = 1 is justified for that definition?
In my understanding the first order Farey series is [0/1, 1/1], and
a zeroth order would either have zero terms or not be well defined -
it would have to consist of terms with denominator 0.

I see the value does match a formula such as a(n) = 1 + Sum_{i=1..n} phi(i);
I'd argue that is coincidental: a(n) = a(n-1) + phi(n) by definition for
n > 1, but not for n = 1.

If my argument seems reasonable, I'd propose only adding a comment to warn
about the relevance of a(0) - I think the entry would be diminished if we
changed the title to be the formula and reduced the Farey connection to
a comment, and it could cause unnecessary problems if we changed the
values of so venerable a sequence.

Hugo


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