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User: Isaac Kaufmann

Isaac Kaufmann has authored 2 sequences.

A333974 Eventual period of A007660 modulo n.

1, 3, 4, 3, 7, 12, 5, 3, 8, 21, 6, 12, 15, 15, 28, 3, 31, 24, 23, 21, 20, 6, 13, 12, 28, 15, 24, 15, 46, 84, 7, 3, 12, 93, 35, 24, 20, 69, 60, 21, 25, 60, 44, 6, 56, 39, 14, 12, 30, 84, 124, 15, 84, 24, 42, 15, 92, 138, 11, 84, 117, 21, 40, 3, 105, 12, 121, 93

Offset: 1

Isaac Kaufmann, Sep 03 2020

Multiplicative: If n = p^x * q^y * ... for distinct primes p,q,... then a(n) = a(p^x) * a(q^y) * ...

			a(1) is trivially 1.
For n=2 the sequence is 0, {0,1,1}, {0,1,1}, hence a(2) = 3.
For n=3 the sequence is 0, {0,1,1,2}, {0,1,1,2}, hence a(3) = 4.
For n=4 the sequence is 0,0,1,1, {2,3,3}, {2,3,3}, hence a(4) = 3.
For n=5 the sequence is 0, {0,1,1,2,3,2,2}, {0,1,1,2,3,2,2}, hence a(5) = 7.

David A. Corneth, PARI program.

See Corneth link \\ David A. Corneth, Sep 04 2020

a(n) | a(k*n), k=2,3,...

More terms from Jinyuan Wang, Sep 04 2020

A308483 Irregular triangle read by rows: T(n,k) = Farey(n,k+1) - Farey(n,k) where Farey(n,k) = A006842(n,k)/A006843(n,k).

Original entry on oeis.org

1, 2, 2, 3, 6, 6, 3, 4, 12, 6, 6, 12, 4, 5, 20, 12, 15, 10, 10, 15, 12, 20, 5, 6, 30, 20, 12, 15, 10, 10, 15, 12, 20, 30, 6, 7, 42, 30, 20, 28, 21, 15, 35, 14, 14, 35, 15, 21, 28, 20, 30, 42, 7, 8, 56, 42, 30, 20, 28, 21, 24, 40, 35, 14, 14, 35, 40, 24, 21, 28, 20, 30, 42, 56, 8

Offset: 1

Isaac Kaufmann, May 30 2019

This is also the product of the denominators of pairs of consecutive terms in the Farey sequence.
Each term of this sequence is an integer: (Proof by induction)
Assume that the reciprocal of Farey differences of order n are the product of the consecutive denominators, i.e., if x/y and c/d are adjacent, then |x/y - c/d| = 1/dy. Let a/b and p/q be adjacent in Farey sequence up to n, such that n+1 = b+q (so only their mediant is in the middle).
As |a/b - p/q| = 1/bq, |aq - bp| = 1, so |aq - bp + ab - ab| = 1, so |a/b - (a+p)/(b+q)| = 1. The base case is trivial. QED

			T(1,1) = 1/(1 - 0);
T(2,1) = 1/(1/2 - 0);
T(2,2) = 1/(1 - 1/2);
T(3,1) = 1/(1/3 - 0);
T(3,2) = 1/(1/2 - 1/3);
T(3,3) = 1/(2/3 - 1/2);
T(3,4) = 1/(1 - 2/3);
...
If written as an array:
  1;
  2,  2;
  3,  6,  6,  3;
  4, 12,  6,  6, 12,  4;
  5, 20, 12, 15, 10, 10, 15, 12, 20, 5;
  ...

Cf. A006842, A006843.

rowf(n) = {my(vf = [0]); for (k=1, n, for (m=1, k, vf = concat(vf, m/k); ); ); vecsort(Set(vf));} \\ A006842/A006843
row(n) = my(vf = rowf(n)); vector(#vf-1, k, 1/(vf[k+1] - vf[k])); \\ Michel Marcus, Jun 07 2019

T(n,k) = Farey(n,k+1) - Farey(n,k) with Farey(n,k) = A006842(n,k)/A006843(n,k).

T(n,k) = A006843(n,k)*A006843(n,k+1).

More terms from Michel Marcus, Jun 07 2019

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User: Isaac Kaufmann

A333974 Eventual period of A007660 modulo n.

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