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A087811
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Numbers n such that ceiling(sqrt(n)) divides n.
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15
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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
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OFFSET
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1,2
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COMMENTS
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Essentially the same as the quarter-squares A002620.
Nonsquare elements 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
A087811(n) is the number of (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
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..300
Wikipedia, Oppermann's conjecture
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
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FORMULA
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a(n) = (n + n mod 2)*(n + 2 - n mod 2)/4.
Numbers of the form k^2 or k^2 - k. - Don Reble, Oct 17 2003
a(1) = 1, a(2) = 2, a(n) = n + a(n - 2). - Alonso del Arte, Jun 18 2005
G.f.: x/((1+x)*(1-x)^3). a(n) = (2*n*(n+2)-(-1)^n+1)/8. - Bruno Berselli, Feb 09 2011
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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(MAGMA) [ n: n in [1..841] | n mod Ceiling(Sqrt(n)) eq 0 ]; // Bruno Berselli, Feb 09 2011
(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
(Haskell)
a087811 n = (n + n `mod` 2) * (n + 2 - n `mod` 2) `div` 4
-- Reinhard Zumkeller, Oct 27 2012
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CROSSREFS
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Cf. A002378, A002620, A003059, A110835, A316841.
Subsequence of A006446.
Sequence in context: A083392 A076921 A002620 * A025699 A224813 A224812
Adjacent sequences: A087808 A087809 A087810 * A087812 A087813 A087814
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KEYWORD
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nonn,easy
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AUTHOR
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Reinhard Zumkeller, Oct 16 2003
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STATUS
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approved
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