A387380 a(n) = number of strict quartets (n,u,v,w) of type 4; see Comments.

0, 0, 0, 0, 1, 6, 3, 8, 7, 18, 8, 35, 12, 32, 41, 45, 19, 67, 23, 85, 73, 69, 31, 139, 62, 87, 95, 145, 46, 210, 52, 155, 142, 128, 163, 281, 69, 150, 176, 310, 79, 342, 85, 272, 322, 198, 95, 449, 176, 326, 259, 345, 114, 451, 309, 496, 303, 269, 132, 753

Offset

1,6

Comments

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

Links

Table of n, a(n) for n=1..60.

Example

For a(5), the unique strict quartet of type 4 is (5,1,2,8).
For a(6), the 6 strict quartets are (6, 1, 2, 13), (6, 1, 3, 7), (6, 2, 3, 5), (6, 3, 2, 7), (6, 4, 3, 1), (6, 5, 2, 1).
For a(7), the 3 strict quartets are (7,1,2,19), (7,1,3,11), (7,3,2,13).
For a(8), the 8 strict quartets are (8,1,2,26), (8,1,4,10), (8,2,3,13), (8,3,2,18), (8,3,4,6), (8,4,2,14), (8,5,2,10), (8,5,4,2).
For a(9), the 7 strict quartets are (9,1,2,34), (9,1,3,21), (9,1,4,14), (9,2,3,18), (9,3,2,25), (9,4,3,12), (9,5,2,16).

Mathematica

ClearAll[solns4];
Options[solns4]={distinct->True};
(*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->True], {u, m-1}]]], {m, 200}]
(* Peter J. C. Moses, Aug 27 2025 *)

Crossrefs

Cf. A385182, A387379.

Sequence in context: A365522 A088246 A179559 * A386754 A086648 A268893

Adjacent sequences: A387377 A387378 A387379 * A387381 A387382 A387383

Keyword

nonn

Author

Clark Kimberling, Sep 03 2025

Status

approved