A368784 a(0) = 1. For n > 0, a(n) is the smallest integer k > n such that (Sum_{i = 1..n} i)/(Sum_{i = n + 1..k} i) < 1/n.

1, 2, 4, 7, 10, 13, 17, 21, 25, 30, 35, 40, 45, 50, 56, 62, 68, 74, 81, 87, 94, 101, 108, 115, 122, 130, 138, 145, 153, 162, 170, 178, 187, 195, 204, 213, 222, 231, 240, 250, 259, 269, 279, 289, 298, 309, 319, 329, 339, 350, 361, 371, 382, 393, 404, 415, 427, 438

Offset

0,2

Comments

(Sum_{i = 1..n} i)/(Sum_{i = n + 1..k} i) = n*(n + 1)/((k - n)*(n + 1 + k)) < 1/n. It follows that k > -1/2 + sqrt(4*n^3 + 8*n^2 + 4*n + 1)/2.

Links

Felix Huber, Table of n, a(n) for n = 0..10000

Formula

a(n) = floor(-1/2 + sqrt(4*n^3 + 8*n^2 + 4*n + 1)/2) + 1.

a(n) = round(sqrt(n*(n+1)^2 + 1/4)). - Chai Wah Wu, Mar 11 2024

Example

a(3) = 7, because (1 + 2 + 3)/(4 + 5 + 6 + 7) = 3/11 < 1/3 and (1 + 2 + 3)/(4 + 5 + 6) = 2/5 > 1/3.

Maple

A368784 := n -> floor(-1/2 + 1/2*sqrt(4*n^3 + 8*n^2 + 4*n + 1)) + 1;
seq(A368784(n), n = 0 .. 57);

Mathematica

a[n_]:= Floor[-1/2 + Sqrt[4*n^3 + 8*n^2 + 4*n + 1]/2] + 1; Array[a, 58, 0] (* Stefano Spezia, Feb 17 2024 *)

Crossrefs

Cf. A003067, A061465, A126022, A130243.

Sequence in context: A003067 A130243 A061465 * A126022 A194172 A173537

Adjacent sequences: A368781 A368782 A368783 * A368785 A368786 A368787

Keyword

nonn,easy

Author

Felix Huber, Feb 15 2024

Status

approved