OFFSET
1,5
COMMENTS
This sequence arises for instance in the following circumstances: let (C_k)_k be the circles with center (0, k) and radius k, k positive integer. n represents time (n >= 0, real for now). For each k, a mobile point M(n, k) (hereafter called k) belongs to C_k. Initially located at (0, 0), k moves counterclockwise at speed 2*Pi. It can be shown that at time n, all k are located on the curve S with polar equation rho = 2*Pi*n*sinc(theta), theta >= 0. When n is a positive integer, a natural and visual classification of points into 3 subsets crops up: the k divisors of n, superposed at (0, 0); the k nondivisors of n such that 1 < k < n, located on the loops of S; the k greater than n, located on the open branch of S. If, for each k in the second subset, one draws the line that passes through (0, 0) and k, then a(n) is the number of distinct lines obtained. In general a(n) is not equal to n - d(n), where d(n) denotes the number of divisors of n, because some k may align. See provided illustration for a(9)=5, section Links. It can be shown that alignment of k1 and k2 occurs iff the condition frac(n/k1) = frac(n/k2) is satisfied.
a(n) = n - 2 iff n is a prime. - Robert G. Wilson v, Jul 21 2018
From Robert G. Wilson v, Jul 26 2018: (Start)
Records: 0, 3, 5, 6, 9, 11, 15, 17, 18, 21, 22, 27, 29, 30, 35, 39, 41, 42, 45, 51, 54, 57, 59, 65, 69, 71, 77, 78, 81, 82, ..., ;
They occur for: 1, 3, 5, 7, 10, 11, 13, 17, 19, 22, 23, 26, 29, 31, 34, 37, 41, 43, 46, 47, 53, 58, 59, 61, 67, 71, 73, 79, ..., ;
First occurrence of k=0,1,2...: 1, 3, 6, 5, 8, 7, 10, 0, 0, 11, 14, 13, 20, 0, 0, 17, 24, 19, 22, 0, 25, 23, 26, 0, 32, 36, ..., ;
Last occurrence of k=0,1,2...: 2, 4, 6, 5, 8, 9, 12, 0, 0, 15, 16, 18, 20, 0, 0, 21, 24, 19, 22, 0, 30, 27, 26, 0, 32, 36, ..., .
(End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..20000
Luc Rousseau, Diagram illustrating a(9)=5
FORMULA
a(p) = p - 2 for any prime p. - Rémy Sigrist, Jul 11 2018
MAPLE
a:= n-> nops({seq(frac(n/k), k=2..n-1)} minus {0}):
seq(a(n), n=1..100); # Alois P. Heinz, Jul 24 2018
MATHEMATICA
a[n_] := Length@ Union@ Mod[n/Range@ n, 1] -1; Array[a, 79] (* Robert G. Wilson v, Jul 21 2018 *)
PROG
(PARI) a(n)={#Set(apply(k->frac(n/k), [1..n]))-1} \\ Andrew Howroyd, Jul 09 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Luc Rousseau, Jul 05 2018
STATUS
approved