A335437 Numbers k with a partition into two distinct parts (s,t) such that k | s*t.

9, 16, 18, 25, 27, 32, 36, 45, 48, 49, 50, 54, 63, 64, 72, 75, 80, 81, 90, 96, 98, 99, 100, 108, 112, 117, 121, 125, 126, 128, 135, 144, 147, 150, 153, 160, 162, 169, 171, 175, 176, 180, 189, 192, 196, 198, 200, 207, 208, 216, 224, 225, 234, 240, 242, 243, 245, 250, 252, 256

Offset: 1

Author

Wesley Ivan Hurt, Jun 10 2020

Keywords

• nonn

Comments

All values of this sequence are nonsquarefree (A013929).

From Peter Munn, Nov 23 2020: (Start)

Numbers whose square part is greater than 4. [Proof follows from s and t having to be multiples of A019554(k), the smallest number whose square is divisible by k.]

Compare with A116451, numbers whose odd part is greater than 3. The self-inverse function A225546(.) maps the members of either one of these sets 1:1 onto the other set.

Compare with A028983, numbers whose squarefree part is greater than 2.

(End)

The asymptotic density of this sequence is 1 - 15/(2*Pi^2). - Amiram Eldar, Mar 08 2021

From Bernard Schott, Jan 09 2022: (Start)

Numbers of the form u*m^2, for u >= 1 and m >= 3 (union of first 2 comments).

A geometric application: in trapezoid ABCD, with AB // CD, the diagonals intersect at E. If the area of triangle ABE is u and the area of triangle CDE is v, with u>v, then the area of trapezoid ABCD is w = u + v + 2*sqrt(u*v); in this case, u, v, w are integer solutions iff (u,v,w) = (k*s^2,k*t^2,k*(s+t)^2), with s>t and k positives; hence, w is a term of this sequence (see IMTS link). (End)

Examples

			16 is in the sequence since it has a partition into two distinct parts (12,4), such that 16 | 12*4 = 48.
		

References

• S. Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 1 of International Mathematical Talent Search, round 7, page 285.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000.
• International Mathematical Talent Search, Problem 1/7, Round 7.
• Eric Weisstein's World of Mathematics, Square part.
• Index to sequences related to Olympiads and other Mathematical Competitions.
• Index entries for sequences related to partitions.

Crossrefs

Complement of A133466 within A013929.

Cf. A019554, A028983, A335234, A335438.

A038838, A046101, A062312\{1}, A195085 are subsequences.

Related to A116451 via A225546.

Programs

Mathematica

Table[If[Sum[(1 - Ceiling[(i*(n - i))/n] + Floor[(i*(n - i))/n]), {i, Floor[(n - 1)/2]}] > 0, n, {}], {n, 300}] // Flatten
f[p_, e_] := p^(2*Floor[e/2]); sqpart[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[256], sqpart[#] > 4 &] (* Amiram Eldar, Mar 08 2021 *)