cp's OEIS Frontend

A385595 The u sequence in quartets (3,u,v,w); i.e., values of u for solutions to 3*(3+u) = v*(v+w), in positive integers, with u,v>=3 and u>=m, sorted by nondecreasing values of u; see Comments.

%I A385595 #8 Jul 11 2025 01:12:56
%S A385595 5,7,9,11,12,13,13,15,17,17,17,18,19,21,21,21,22,23,25,25,25,27,27,27,
%T A385595 29,29,29,30,31,32,32,33,33,33,35,36,37,37,37,37,37,39,39,39,41,41,41,
%U A385595 42,42,43,45,45,45,45,46,47,47,47,48,49,49,49,51,51,52
%N A385595 The u sequence in quartets (3,u,v,w); i.e., values of u for solutions to 3*(3+u) = v*(v+w), in positive integers, with u,v>=3 and u>=m, sorted by nondecreasing values of u; see Comments.
%C A385595 A 4-tuple (m,u,v,w) is a quartet if m,u,v,w are positive integers such that m<=u, m<v, and m*(m+u) = v*(v+w), with the values of u in nondecreasing order. When there is more than one solution for given m and u, the values of v are arranged in increasing order. Here, m=3; for m=1, see A385182.
%e A385595 First 30 quartets (3,u,v,w):
%e A385595    m    u    v    w
%e A385595    3    5    4    2
%e A385595    3    7    5    1
%e A385595    3    9    4    5
%e A385595    3   11    6    1
%e A385595    3   12    5    4
%e A385595    3   13    4    8
%e A385595    3   13    6    2
%e A385595    3   15    6    3
%e A385595    3   17    4   11
%e A385595    3   17    5    7
%e A385595    3   17    6    4
%e A385595    3   18    7    2
%e A385595    3   19    6    5
%e A385595    3   21    4   14
%e A385595    3   21    4   14
%e A385595    3   21    6    6
%e A385595    3   21    8    1
%e A385595    3   22    5   10
%e A385595    3   23    6    7
%e A385595    3   25    4   17
%e A385595    3   25    6    8
%e A385595    3   25    7    5
%e A385595    3   27    5   13
%e A385595    3   27    6    9
%e A385595    3   27    9    1
%e A385595    3   29    4   20
%e A385595    3   29    6   10
%e A385595    3   29    8    4
%e A385595    3   30    9    2
%e A385595    3   31    6   11
%e A385595 3(3+13) = 4(4+8) = 6(6+2), so (3,13,4,8) and (3,13,6,2) are rows.
%t A385595 Clear[solnsM];
%t A385595 solnsM[m_, max_] := Module[{ans = {}, rhs = {}, u, v, w, lhs, matching},
%t A385595 Do[Do[AppendTo[rhs, {v*(v + w), v, w}], {w, max}], {v, m*(m + max)}];
%t A385595 rhs = GatherBy[rhs, First];
%t A385595 Do[lhs = m*(m + u); matching = Select[rhs, #[[1, 1]] == lhs &];
%t A385595 If[Length[matching] > 0, Do[AppendTo[ans,
%t A385595 Map[{m, u, #[[2]], #[[3]]} &, matching[[1]]]], {i,
%t A385595 Length[matching]}]], {u, max}];
%t A385595 ans = Flatten[ans, 1];
%t A385595 Select[Union[Map[Sort[{#, RotateLeft[#, 2]}][[1]] &,
%t A385595 Sort[Select[DeleteDuplicates[
%t A385595 ans], {#[[1]], #[[2]]} =!= {#[[3]], #[[4]]} &]]]], #[[1]] == m &]];
%t A385595 TableForm[solns = solnsM[3, 140], TableHeadings -> {None, {"m", "u", "v", "w"}}]
%t A385595 aa = Flatten[solns]
%t A385595 Map[#[[2]] &, solns]    (* u, A385595 *)
%t A385595 Map[#[[3]] &, solns]    (* v, A385596 *)
%t A385595 Map[#[[4]] &, solns]    (* w, A385597 *)
%t A385595 (*_Peter J.C.Moses_, Jun 15 2025*)
%Y A385595 Cf. A385182, A385592, A385596, A385597.
%K A385595 nonn
%O A385595 1,1
%A A385595 _Clark Kimberling_, Jul 07 2025