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A087811 Numbers k such that ceiling(sqrt(k)) divides k. 18

%I #156 Jan 01 2024 13:30:22

%S 1,2,4,6,9,12,16,20,25,30,36,42,49,56,64,72,81,90,100,110,121,132,144,

%T 156,169,182,196,210,225,240,256,272,289,306,324,342,361,380,400,420,

%U 441,462,484,506,529,552,576,600,625,650,676,702,729,756,784,812,841

%N Numbers k such that ceiling(sqrt(k)) divides k.

%C Essentially the same as the quarter-squares A002620.

%C Nonsquare terms of this sequence are given by A002378. - _Max Alekseyev_, Nov 27 2006

%C 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

%C 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

%C 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

%C 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

%C 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

%C Also number of pairs (x,y) with 0 < x <= y <= n, x + y > n. - _Ralf Steiner_, Jan 05 2020

%H Vincenzo Librandi, <a href="/A087811/b087811.txt">Table of n, a(n) for n = 1..300</a>

%H Ya-Ping Lu, <a href="/A087811/a087811.pdf">Illustration of the Terms in A087811 on the Square Spiral</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Oppermann%27s_conjecture">Oppermann's conjecture</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).

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

%F Numbers of the form m^2 or m^2 - m. - _Don Reble_, Oct 17 2003

%F a(1) = 1, a(2) = 2, a(n) = n + a(n - 2). - _Alonso del Arte_, Jun 18 2005

%F From _Bruno Berselli_, Feb 09 2011: (Start)

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

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

%F 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

%F a(n) = (C(n+2,2) - floor((n+2)/2))/2. - _Mircea Merca_, Nov 23 2013

%F a(n) = ((-1)^n*(-1 + (-1)^n*(1 + 2*n*(2 + n))))/8. - _Fred Daniel Kline_, Jan 06 2015

%F a(n) = Product_{k=1...n-1} (1 + 2 / (k + k mod 2)), n >= 1. - _Fred Daniel Kline_, Oct 30 2016

%F E.g.f.: (1/4)*(x*(3 + x)*cosh(x) + (1 + 3*x + x^2)*sinh(x)). - _Stefano Spezia_, Jan 05 2020

%F a(n) = (n*(n+2)+(n mod 2))/4. - _Chai Wah Wu_, Jul 27 2022

%F Sum_{n>=1} 1/a(n) = Pi^2/6 + 1. - _Amiram Eldar_, Sep 17 2022

%F a(n) = A024206(n) + 1. - _Ya-Ping Lu_, Dec 29 2023

%p f:= gfun:-rectoproc({a(n)=n+a(n-2),a(1)=1,a(2)=2},a(n),remember):

%p map(f, [$1..100]); # _Robert Israel_, Aug 31 2016

%t a[1] := 1; a[2] := 2; a[n_] := n + a[n - 2]; Table[a[n], {n, 57}] (* _Alonso del Arte_ *)

%t GaloisNumber[n_, q_] :=

%t Sum[QBinomial[n, m, q], {m, 0, n}]; aa = {}; Do[

%t sub = Table[GaloisNumber[m, n], {n, 0, 200}];

%t pp = InterpolatingPolynomial[sub, x]; pol = pp /. x -> n + 1;

%t coef = CoefficientList[pol, n];

%t AppendTo[aa, Length[coef] - 1], {m, 2, 25}]; aa (* _Artur Jasinski_, Aug 31 2016 *)

%t Select[Range[900],Divisible[#,Ceiling[Sqrt[#]]]&] (* or *) LinearRecurrence[ {2,0,-2,1},{1,2,4,6},60] (* _Harvey P. Dale_, Nov 06 2016 *)

%o (Magma) [ n: n in [1..841] | n mod Ceiling(Sqrt(n)) eq 0 ]; // _Bruno Berselli_, Feb 09 2011

%o (PARI) a(n)=(n+n%2)*(n+2-n%2)/4 \\ _Charles R Greathouse IV_, Apr 03 2012

%o (PARI) j=0;for(k=1,850,s=sqrtint(4*k+1);if(s>j,j=s;print1(k,", "))) \\ _Hugo Pfoertner_, Sep 17 2018

%o (Haskell)

%o a087811 n = (n + n `mod` 2) * (n + 2 - n `mod` 2) `div` 4

%o -- _Reinhard Zumkeller_, Oct 27 2012

%o (Python)

%o def A087811(n): return n*(n+2)+(n&1)>>2 # _Chai Wah Wu_, Jul 27 2022

%Y Cf. A002378, A002620, A003059, A024206, A110835, A316841.

%Y Subsequence of A006446.

%K nonn,easy

%O 1,2

%A _Reinhard Zumkeller_, Oct 16 2003

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Last modified March 29 08:13 EDT 2024. Contains 371265 sequences. (Running on oeis4.)