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

%I A335437 #55 Feb 16 2025 08:34:00
%S A335437 9,16,18,25,27,32,36,45,48,49,50,54,63,64,72,75,80,81,90,96,98,99,100,
%T A335437 108,112,117,121,125,126,128,135,144,147,150,153,160,162,169,171,175,
%U A335437 176,180,189,192,196,198,200,207,208,216,224,225,234,240,242,243,245,250,252,256
%N A335437 Numbers k with a partition into two distinct parts (s,t) such that k | s*t.
%C A335437 All values of this sequence are nonsquarefree (A013929).
%C A335437 From _Peter Munn_, Nov 23 2020: (Start)
%C A335437 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.]
%C A335437 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.
%C A335437 Compare with A028983, numbers whose squarefree part is greater than 2.
%C A335437 (End)
%C A335437 The asymptotic density of this sequence is 1 - 15/(2*Pi^2). - _Amiram Eldar_, Mar 08 2021
%C A335437 From _Bernard Schott_, Jan 09 2022: (Start)
%C A335437 Numbers of the form u*m^2, for u >= 1 and m >= 3 (union of first 2 comments).
%C A335437 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)
%D A335437 S. Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 1 of International Mathematical Talent Search, round 7, page 285.
%H A335437 Amiram Eldar, <a href="/A335437/b335437.txt">Table of n, a(n) for n = 1..10000</a>.
%H A335437 International Mathematical Talent Search, <a href="https://www2.cms.math.ca/Competitions/IMTS/imts7.html">Problem 1/7</a>, Round 7.
%H A335437 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquarePart.html">Square part</a>.
%H A335437 <a href="/index/O#Olympiads">Index to sequences related to Olympiads and other Mathematical Competitions</a>.
%H A335437 <a href="/index/Par#part">Index entries for sequences related to partitions</a>.
%e A335437 16 is in the sequence since it has a partition into two distinct parts (12,4), such that 16 | 12*4 = 48.
%t A335437 Table[If[Sum[(1 - Ceiling[(i*(n - i))/n] + Floor[(i*(n - i))/n]), {i, Floor[(n - 1)/2]}] > 0, n, {}], {n, 300}] // Flatten
%t A335437 f[p_, e_] := p^(2*Floor[e/2]); sqpart[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[256], sqpart[#] > 4 &] (* _Amiram Eldar_, Mar 08 2021 *)
%Y A335437 Complement of A133466 within A013929.
%Y A335437 Cf. A019554, A028983, A335234, A335438.
%Y A335437 A038838, A046101, A062312\{1}, A195085 are subsequences.
%Y A335437 Related to A116451 via A225546.
%K A335437 nonn
%O A335437 1,1
%A A335437 _Wesley Ivan Hurt_, Jun 10 2020