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A360452
Number of fractions c/d with |c| <= d <= 2n and odd denominator when factors of 2 are canceled.
1
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.
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.
(Python)
# uses programs from A002088 and A049690
def A360452(n): return (A002088((n<<1)-1)-A049690(n-1)<<1)|1 if n else 0 # Chai Wah Wu, Aug 04 2024
CROSSREFS
Sequence in context: A324719 A170884 A182836 * A097080 A274008 A146742
KEYWORD
nonn,easy
AUTHOR
M. F. Hasler, Mar 26 2023
STATUS
approved