A360452 Number of fractions c/d with |c| <= d <= 2n and odd denominator when factors of 2 are canceled.

0, 3, 7, 15, 27, 39, 59, 83, 99, 131, 167, 191, 235, 275, 311, 367, 427, 467, 515, 587, 635, 715, 799, 847, 939, 1023, 1087, 1191, 1271, 1343, 1459, 1579, 1651, 1747, 1879, 1967, 2107, 2251, 2331, 2451, 2607, 2715, 2879, 3007, 3119, 3295, 3439, 3559, 3703, 3895, 4015

Offset

0,2

Comments

Using d <= 2n or d <= 2n-1 gives the same result, therefore we use 2n and not just n for the upper limit of the denominator. Indeed, using an even d will only yield the same simplified fractions with odd denominators as one gets for d/2.

Links

Table of n, a(n) for n=0..50.

Formula

a(n) = 2*A099957(n)+1 for n > 0.

Example

For n = 0, there is no possible fraction, since the denominator can't be zero.
For n = 1, we have a(1) = #{ -1/1, 0/1, 1/1} = 3; using denominator d = 2 would not give other elements with odd denominator after cancellations, cf. comments.
For n = 2, we have a(2) = #{-1/1, -2/3, -1/3, 0, 1/3, 2/3, 1/1} = 7.
For n = 3, we have a(3) = #{-1/1, -4/5, -2/3, -3/5, -2/5, -1/3, -1/5, 0, 1/5, 1/3, 2/5, 3/5, 2/3, 4/5, 1/1} = 15. As explained in comments, only odd d are useful.

Prog

(PARI) a(n)=#Set(concat([[c/d|c<-[-d..d], d && denominator(c/d)%2]|d<-[0..n*2]])) \\ For illustration only. Remove the # to see the elements. Obviously the code could be optimized.
(PARI) apply( {A360452(n) = sum(i=0, n-1, eulerphi(2*i+1))*2+!!n}, [0..10]) \\ This should be used to define the "official" function A360452.

Crossrefs

Cf. A099957, A049691.

Sequence in context: A324719 A170884 A182836 * A097080 A274008 A146742

Adjacent sequences: A360449 A360450 A360451 * A360453 A360454 A360455

Keyword

nonn,easy

Author

M. F. Hasler, Mar 26 2023

Status

approved