A226690 Irregular table read by rows: T(m,n) = Numerators of minimal distance between m elements of the Farey sequence F_n.

0, 1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 5, 1, 3, 1, 0, 1, 2, 1, 3, 2, 7, 11, 3, 4, 1, 0, 1, 1, 1, 7, 1, 13, 1, 7, 19, 2, 5, 1, 0, 1, 2, 3, 1, 4, 9, 2, 5, 3, 16, 11, 4, 17, 23, 29, 5, 6, 1, 0, 1, 1, 3, 1, 9, 5, 1, 11, 17, 3, 25, 19, 1, 13, 33, 5, 27, 17, 41, 3, 7, 1, 0, 1, 2

Offset

1,9

Comments

The first element of each row is 0 (because the distance from any element to itself is 0), the last element of each row is 1 (because the distance from 0 to 1 is 1).

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Example

Table begins:
0, 1;
0, 1/2, 1;
0, 1/6, 1/3, 2/3, 1;
0, 1/12, 1/4, 5/12, 1/2, 3/4, 1;
Consider the third row. The Farey series for 3 is (0, 1/3, 1/2, 2/3, 1). 0 is always first. The closest two elements are 1/3 and 1/2, so 1/6 is next. The closest three are 1/3 and 2/3 with distance 1/3. Including either endpoint gives distance 2/3; both gives distance 1.

Prog

(PARI) Farey(n)=my(v=List([0, 1])); for(a=2, n, for(b=1, a-1, listput(v, b/a))); vecsort(Vec(v), , 8)
f(n)=my(F=Farey(n)); vector(#F, k, my(b=1); for(i=k, #F, b=min(F[i]-F[i-k+1], b)); b)
for(n=1, 9, v=apply(numerator, f(n)); for(i=1, #v, print1(v[i]", ")))

Crossrefs

Cf. A226691.

Sequence in context: A355576 A198062 A347617 * A318557 A245683 A273712

Adjacent sequences: A226687 A226688 A226689 * A226691 A226692 A226693

Keyword

nonn,tabf

Author

Charles R Greathouse IV, Jun 15 2013

Status

approved