A024584 -id:A024584 - cp's OEIS frontend

A024573 a(n) = floor(1/frac(n*e)).

1, 2, 6, 1, 1, 3, 35, 1, 2, 5, 1, 1, 2, 17, 1, 2, 4, 1, 1, 2, 11, 1, 1, 4, 1, 1, 2, 8, 1, 1, 3, 1, 1, 2, 7, 1, 1, 3, 76, 1, 2, 5, 1, 1, 3, 24, 1, 2, 5, 1, 1, 2, 14, 1, 1, 4, 1, 1, 2, 10, 1, 1, 3, 1, 1, 2, 8, 1, 1, 3, 1, 1, 2, 6, 1, 1, 3, 38, 1, 2, 5, 1, 1, 2, 18, 1, 2, 4, 1, 1, 2, 12, 1, 1, 4, 1, 1, 2, 9, 1

Offset: 1

Clark Kimberling

nonn

From Hieronymus Fischer, Apr 15 2012: (Start)
The sequence is well defined, since frac(n*e)>0 for n>0.
Let b(n,m) = |{a(k)| 1<=k<=n, a(k)>=m}| be the number of the first n terms which are >= m >= 1. Then, lim b(n,m)/n = 1/m for n-->oo since frac(n*e) is uniformly distributed. (End)

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A024584, A022843, A000142, A000027.

Cf. A190860, A190847.

seq(floor(1/frac(n*exp(1))), n=1..30); # Ridouane Oudra, Jun 09 2025

f[n_] := Floor[1/FractionalPart[n*E]]; Array[f, 100] (* Robert G. Wilson v, Apr 17 2012 *)

From Ridouane Oudra, Jun 09 2025: (Start)
a(n!) = n.
a(n) = 1 if n is in A190860.
a(n) > 1 if n is in A190847. (End)

A024585 a(n) = Sum_{k=1..n} [ 1/{k*Pi} ], where {x} := x - [ x ].

Original entry on oeis.org

7, 10, 12, 13, 14, 15, 16, 23, 26, 28, 29, 30, 31, 32, 40, 43, 45, 46, 47, 48, 49, 57, 60, 62, 63, 64, 65, 66, 75, 79, 81, 82, 83, 84, 85, 95, 99, 101, 102, 103, 104, 105, 116, 120, 122, 123, 124, 125, 126, 138, 142, 144, 145, 146, 147, 148, 162, 166, 168, 170, 171, 172, 173, 189

Offset: 1

Clark Kimberling

nonn

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A024586. Partial sums of A024584.

Table[Sum[Floor[1/FractionalPart[k*Pi]], {k, n}], {n, 100}] (* Clark Kimberling, Aug 18 2012 *)

a(n) = sum(k=1, n, floor(1/frac((k*Pi)))); \\ Michel Marcus, Jul 17 2019

A024583 a(n) = floor(n/{n*Pi}), where { } = fractional part.

Original entry on oeis.org

7, 7, 7, 7, 7, 7, 7, 60, 32, 24, 19, 17, 15, 14, 121, 60, 41, 32, 27, 24, 21, 191, 89, 60, 46, 38, 32, 29, 273, 121, 79, 60, 49, 41, 36, 369, 154, 99, 74, 60, 50, 44, 485, 191, 121, 89, 71, 60, 52, 627, 230, 143, 105, 83, 69, 60, 805, 273, 166, 121, 95, 79, 68, 1033, 319, 191, 137, 108

Offset: 1

Clark Kimberling

nonn

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A024584.

Table[Floor[n/FractionalPart[n*Pi]], {n, 70}] (* Clark Kimberling, Aug 18 2012 *)

cp's OEIS Frontend

A024573 a(n) = floor(1/frac(n*e)).

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A024585 a(n) = Sum_{k=1..n} [ 1/{k*Pi} ], where {x} := x - [ x ].

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A024583 a(n) = floor(n/{n*Pi}), where { } = fractional part.

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