A033501 - OEIS

%I #10 Sep 23 2017 19:23:21

%S 1,2,3,4,6,8,9,12,15,16,18,20,24,25,28,30,35,36,40,42,48,49,54,56,60,

%T 63,64,70,72,77,80,81,88,90,96,99,100,108,110,117,120,121,130,132,140,

%U 143,144,150,154,156,165,168,169,176,180,182,192,195,196,204,208,210

%N Almost-squares: m such that m/p(m) >= k/p(k) for all k<m, where p(m) is the least perimeter of a rectangle with integer side lengths and area m.

%C Also integers that can be written in the form k*(k+h), for some integers k>=1 and 0 <= h <= T(k), where T(x) is the number of triangular numbers binomial(x+1,2) not exceeding x. (Corollary 1 in Greg Martin's article) - _Hugo Pfoertner_, Sep 23 2017

%H Hugo Pfoertner, <a href="/A033501/b033501.txt">Table of n, a(n) for n = 1..600</a>

%H Greg Martin, <a href="http://arXiv.org/abs/math.NT/9807108">Farmer Ted Goes Natural</a>, Math. Mag. 72 (1999), no. 4, 259-276.

%H Hugo Pfoertner, <a href="/A033501/a033501.pdf">Plot of R(x).</a> R(x)=A(x)-x^(3/4)*2*sqrt(2)/3-sqrt(x)/2, where A(x) is the number of almost-squares not exceeding x.

%t chs={1}; For[ n=2, n<=99, n++, chs=Join[ chs, Reverse[ Table[ (n-1-i)(n+i), {i, 0, (Sqrt[ 2n-1 ]-1)/2} ] ], Reverse[ Table[ (n-i)(n+i), {i, 0, n/Sqrt[ 2n-1 ]} ] ] ] ]

%t (*code uses alternate characterization, lists almost-squares up to 99^2*)

%Y Cf. A000217.

%K nonn,easy

%O 1,2

%A _Greg Martin_; suggested by Jon Grantham.