OFFSET
1,1
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)
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.
EXAMPLE
16 is in the sequence since it has a partition into two distinct parts (12,4), such that 16 | 12*4 = 48.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 10 2020
STATUS
approved