A316519 a(n) is the number of distinct nonzero values frac(n/k) can take for k in 1..n, where frac denotes the fractional part.

0, 0, 1, 1, 3, 2, 5, 4, 5, 6, 9, 6, 11, 10, 9, 10, 15, 11, 17, 12, 15, 18, 21, 16, 20, 22, 21, 20, 27, 20, 29, 24, 27, 30, 27, 25, 35, 34, 33, 28, 39, 30, 41, 36, 33, 42, 45, 35, 43, 40, 45, 44, 51, 43, 47, 43, 51, 54, 57, 44, 59, 58, 51, 53, 56, 54, 65, 60, 63, 55, 69, 53, 71, 70, 63, 68, 67, 66, 77

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

a(n) >= A049820(n) = n - d(n) where d is A000005. - Robert G. Wilson v, Jul 22 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

Cf. A049820, A000005.

Sequence in context: A259846 A087669 A053087 * A062327 A075491 A326730

Adjacent sequences: A316516 A316517 A316518 * A316520 A316521 A316522

Keyword

nonn

Author

Luc Rousseau, Jul 05 2018

Status

approved