A033501 Almost-squares: m such that m/p(m) >= k/p(k) for all k

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

Author

Greg Martin; suggested by Jon Grantham

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A000217.

Programs

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*)