A033501 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.

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, 63, 64, 70, 72, 77, 80, 81, 88, 90, 96, 99, 100, 108, 110, 117, 120, 121, 130, 132, 140, 143, 144, 150, 154, 156, 165, 168, 169, 176, 180, 182, 192, 195, 196, 204, 208, 210

Offset

1,2

Comments

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

Links

Hugo Pfoertner, Table of n, a(n) for n = 1..600

Greg Martin, Farmer Ted Goes Natural, Math. Mag. 72 (1999), no. 4, 259-276.

Hugo Pfoertner, Plot of R(x). 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.

Mathematica

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

Crossrefs

Cf. A000217.

Sequence in context: A145807 A278962 A122380 * A336504 A331827 A231404

Adjacent sequences: A033498 A033499 A033500 * A033502 A033503 A033504

Keyword

nonn,easy

Author

Greg Martin; suggested by Jon Grantham.

Status

approved