A087811 Numbers k such that ceiling(sqrt(k)) divides k.

1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100, 110, 121, 132, 144, 156, 169, 182, 196, 210, 225, 240, 256, 272, 289, 306, 324, 342, 361, 380, 400, 420, 441, 462, 484, 506, 529, 552, 576, 600, 625, 650, 676, 702, 729, 756, 784, 812, 841

Offset: 1

Author

Reinhard Zumkeller, Oct 16 2003

Keywords

• nonn
• easy

Comments

Essentially the same as the quarter-squares A002620.

Nonsquare terms of this sequence are given by A002378. - Max Alekseyev, Nov 27 2006

This also gives the number of ways to make change for "c" cents using only pennies, nickels and dimes. You must first set n=floor(c/5), to account for the 5-repetitive nature of the task. - Adam Sasson, Feb 09 2011

These are the segment boundaries of Oppermann's conjecture (1882): n^2-n < p < n^2 < p < n^2+n. - Fred Daniel Kline, Apr 07 2011

a(n) is the number of triples (w,x,y) having all terms in {0..n} and w=2*x+y. - Clark Kimberling, Jun 04 2012

a(n+1) is also the number of points with integer coordinates inside a rectangle isosceles triangle with hypotenuse [0,n] (see A115065 for an equilateral triangle). - Michel Marcus, Aug 05 2013

a(n) = degree of generating polynomials of Galois numbers in (n+1)-dimensional vector space, defined as total number of subspaces in (n+1) space over GF(n) (see Mathematica procedure), when n is a power of a prime. - Artur Jasinski, Aug 31 2016, corrected by Robert Israel, Sep 23 2016

Also number of pairs (x,y) with 0 < x <= y <= n, x + y > n. - Ralf Steiner, Jan 05 2020

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..300
• Ya-Ping Lu, Illustration of the Terms in A087811 on the Square Spiral
• Wikipedia, Oppermann's conjecture.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A002378, A002620, A003059, A024206, A110835, A316841, A059260.

Subsequence of A006446.

Programs

Haskell

a087811 n = (n + n `mod` 2) * (n + 2 - n `mod` 2) `div` 4
-- Reinhard Zumkeller, Oct 27 2012

Magma

[ n: n in [1..841] | n mod Ceiling(Sqrt(n)) eq 0 ]; // Bruno Berselli, Feb 09 2011

Maple

f:= gfun:-rectoproc({a(n)=n+a(n-2),a(1)=1,a(2)=2},a(n),remember):
map(f, [$1..100]); # Robert Israel, Aug 31 2016

Mathematica

a[1] := 1; a[2] := 2; a[n_] := n + a[n - 2]; Table[a[n], {n, 57}] (* Alonso del Arte *)
GaloisNumber[n_, q_] :=
Sum[QBinomial[n, m, q], {m, 0, n}]; aa = {}; Do[
sub = Table[GaloisNumber[m, n], {n, 0, 200}];
pp = InterpolatingPolynomial[sub, x]; pol = pp /. x -> n + 1;
coef = CoefficientList[pol, n];
AppendTo[aa, Length[coef] - 1], {m, 2, 25}]; aa (* Artur Jasinski, Aug 31 2016 *)
Select[Range[900],Divisible[#,Ceiling[Sqrt[#]]]&] (* or *) LinearRecurrence[ {2,0,-2,1},{1,2,4,6},60] (* Harvey P. Dale, Nov 06 2016 *)

PARI

a(n)=(n+n%2)*(n+2-n%2)/4 \\ Charles R Greathouse IV, Apr 03 2012

PARI

j=0;for(k=1,850,s=sqrtint(4*k+1);if(s>j,j=s;print1(k,", "))) \\ Hugo Pfoertner, Sep 17 2018

Python

def A087811(n): return n*(n+2)+(n&1)>>2 # Chai Wah Wu, Jul 27 2022

Formula

a(n) = (n + n mod 2)*(n + 2 - n mod 2)/4.

Numbers of the form m^2 or m^2 - m. - Don Reble, Oct 17 2003

a(1) = 1, a(2) = 2, a(n) = n + a(n - 2). - Alonso del Arte, Jun 18 2005

From Bruno Berselli, Feb 09 2011: (Start)

G.f.: x/((1+x)*(1-x)^3).

a(n) = (2*n*(n+2)-(-1)^n+1)/8. (End)

G.f.: G(0)/(2*(1-x^2)*(1-x)), where G(k) = 1 + 1/(1 - x*(2*k+1)/(x*(2*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013

a(n) = (C(n+2,2) - floor((n+2)/2))/2. - Mircea Merca, Nov 23 2013

a(n) = ((-1)^n*(-1 + (-1)^n*(1 + 2*n*(2 + n))))/8. - Fred Daniel Kline, Jan 06 2015

a(n) = Product_{k=1...n-1} (1 + 2 / (k + k mod 2)), n >= 1. - Fred Daniel Kline, Oct 30 2016

E.g.f.: (1/4)*(x*(3 + x)*cosh(x) + (1 + 3*x + x^2)*sinh(x)). - Stefano Spezia, Jan 05 2020

a(n) = (n*(n+2)+(n mod 2))/4. - Chai Wah Wu, Jul 27 2022

Sum_{n>=1} 1/a(n) = Pi^2/6 + 1. - Amiram Eldar, Sep 17 2022

a(n) = A024206(n) + 1. - Ya-Ping Lu, Dec 29 2023