A038759 a(n) = ceiling(sqrt(n))*floor(sqrt(n)).

0, 1, 2, 2, 4, 6, 6, 6, 6, 9, 12, 12, 12, 12, 12, 12, 16, 20, 20, 20, 20, 20, 20, 20, 20, 25, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 36, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 49, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 64, 72, 72, 72, 72, 72, 72, 72

Offset

0,3

Comments

a(n) = n iff n is a square or a pronic (or heteromecic) number of form k(k+1). The sequence interleaves individual squares with 2k copies of each pronic.

Links

Table of n, a(n) for n=0..71.

Formula

a(n) = A003059(n)*A000196(n) = n - A038760(n).

a(A002620(n)) = A002620(n). - Bernard Schott, Nov 06 2022

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 (A072691). - Amiram Eldar, Dec 04 2022

Example

a(31) = 30 since 6 and 5 are on either side of the square root of 31 and 6*5 = 30.

Mathematica

a[n_] := Ceiling[Sqrt[n]]*Floor[Sqrt[n]]; Array[a, 70, 0] (* Amiram Eldar, Dec 04 2022 *)

Prog

(Python)
from math import isqrt
def A038759(n): return m+n+k if (m:=(k:=isqrt(n))**2-n) else n # Chai Wah Wu, Jul 28 2022
(PARI) a(n) = my(r, s=sqrtint(n, &r)); if(r, n-r+s, n); \\ Kevin Ryde, Jul 30 2022

Crossrefs

Cf. A000196, A002378, A002620, A003059, A038760, A053187, A072691.

Sequence in context: A118960 A107797 A316788 * A045999 A075569 A338276

Adjacent sequences: A038756 A038757 A038758 * A038760 A038761 A038762

Keyword

nonn,easy

Author

Henry Bottomley, May 03 2000

Status

approved