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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 17 16:34 EDT 2021. Contains 348065 sequences. (Running on oeis4.)