A387379 a(n) = number of quartets (n,u,v,w) of type 4; see Comments.

0, 0, 1, 3, 4, 11, 8, 17, 18, 28, 17, 49, 23, 46, 57, 64, 34, 91, 40, 108, 95, 91, 52, 169, 92, 113, 125, 177, 73, 247, 81, 194, 176, 162, 198, 329, 104, 188, 216, 356, 118, 391, 126, 320, 374, 244, 140, 507, 232, 388, 311, 401, 165, 514, 365, 559, 361, 327

Offset

1,4

Comments

A 4-tuple (n,u,v,w) is a quartet of type 4 if n,u,v,w are positive integers such that n > v > 1 and n*(n - u) = v*(v + w).

Links

Table of n, a(n) for n=1..58.

Example

For n=3, the unique quartet of type 4 is (3,1,2,1).
For n=4, the 3 quartets are (4,1,2,4), (4,1,3,1), (4,2,2,2).
For n=5, the 4 quartets are (5,1,2,8), (5,1,4,1), (5,2,3,2), (5,3,2,3).
For n=6, the 11 quartets are (6,1,2,13), (6,1,3,7), (6,1,5,1), (6,2,2,10), (6,2,3,5), (6,2,4,2), (6,3,2,7), (6,3,3,3), (6,4,2,4), (6,4,3,1), (6,5,2,1).
For n=7, the 8 quartets are (7,1,2,19), (7,1,3,11), (7,1,6,1), (7,2,5,2), (7,3,2,12), (7,3,4,3), (7,4,3,4), (7,5,2,5)

Mathematica

ClearAll[solns4];
Options[solns4]={distinct->False};
(*types:1:m(m+u)=v(v+w) 2:m(m+u)=v(v-w) 3:m(m-u)=v(v-w) 4:m(m-u)=v(v+w), v>1*)
solns4[m_, u_, type:(1|2|3|4), OptionsPattern[]]:=Module[{sU, sW, n, nn, sgn, divs, tups, unique},
{sU, sW}=Switch[type, 1, {1, 1}, 2, {1, -1}, 3, {-1, -1}, 4, {-1, 1}];
n=m*(m+sU*u);
tups=If[sW==1, If[n<=0, {}, divs=Select[Divisors[n], #>n/#&]; (Map[{m, u, n/#, #-n/#}&, divs])], If[n==0, {}, nn=Abs[n]; sgn=Sign[n];
divs=Divisors[nn];
If[sgn>0, divs=Select[divs, #<nn/#&]]; ({m, u, nn/#, nn/#-sgn #}&/@divs)]];
unique=TrueQ[OptionValue[distinct]];
Select[tups, #[[3]]>1&&#[[4]]>0&&(!unique||Length@DeleteDuplicates[#]==4)&]];
Table[Total[Map[Length, Table[solns4[m, u, 4, distinct->False], {u, m-1}]]], {m, 200}]
(* Peter J. C. Moses, Aug 27 2025 *)

Crossrefs

Cf. A385182, A387380.

Sequence in context: A220848 A370605 A232891 * A096223 A232862 A344460

Adjacent sequences: A387376 A387377 A387378 * A387380 A387381 A387382

Keyword

nonn

Author

Clark Kimberling, Sep 03 2025

Status

approved