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

Author

Henry Bottomley, May 03 2000

Keywords

• nonn
• easy

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.

Examples

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

Crossrefs

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

Programs

Mathematica

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

PARI

a(n) = my(r,s=sqrtint(n,&r)); if(r, n-r+s, n); \\ Kevin Ryde, Jul 30 2022

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

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