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A385182 Values of u in the quartets (1,u,v,w); i.e., values of u for solutions to (1+u) = v*(v+w), in positive integers, with v>1, sorted by nondecreasing values of u; see Comments.

%I A385182 #27 Jul 10 2025 00:14:03
%S A385182 5,7,9,11,11,13,14,15,17,17,19,19,20,21,23,23,23,25,26,27,27,29,29,29,
%T A385182 31,31,32,33,34,35,35,35,37,38,39,39,39,41,41,41,43,43,44,44,45,47,47,
%U A385182 47,47,49,49,50,51,51,53,53,53,54,55,55,55,56,57,59,59
%N A385182 Values of u in the quartets (1,u,v,w); i.e., values of u for solutions to (1+u) = v*(v+w), in positive integers, with v>1, sorted by nondecreasing values of u; see Comments.
%C A385182 A 4-tuple (m,u,v,w) is a quartet if m,u,v,w are positive integers such that 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=1.
%C A385182 If m is a prime, then (u,v,w) = (m+2,m+1,m-1) is the first solution (in the defined ordering of triples).
%C A385182 u >= 1 appears A056924(u+1)-1 times. - _Pontus von Brömssen_, Jul 06 2025
%e A385182 First 30 quartets (1,u,v,w):
%e A385182   m    u    v    w
%e A385182   1    5    2    1
%e A385182   1    7    2    2
%e A385182   1    9    2    3
%e A385182   1   11    2    4
%e A385182   1   11    3    1
%e A385182   1   13    2    5
%e A385182   1   14    3    2
%e A385182   1   15    2    6
%e A385182   1   17    2    7
%e A385182   1   17    3    3
%e A385182   1   19    2    8
%e A385182   1   19    4    1
%e A385182   1   20    3    4
%e A385182   1   21    2    9
%e A385182   1   23    2   10
%e A385182   1   23    3    5
%e A385182   1   23    4    2
%e A385182   1   25    2   11
%e A385182   1   26    3    6
%e A385182   1   27    2   12
%e A385182   1   27    4    3
%e A385182   1   29    2   13
%e A385182   1   29    3    7
%e A385182   1   29    5    1
%e A385182   1   31    2   14
%e A385182   1   31    4    4
%e A385182   1   32    3    8
%e A385182   1   33    2   15
%e A385182   1   34    5    2
%e A385182   1   35    2   16
%e A385182 1*(1+23) = 2*(2+10) = 3*(3+5) = 4*(4+2), so three of the rows are (1,23,2,10), (1,23,3,5), and (1,23,4,2).
%t A385182 Clear[solnsM];
%t A385182 solnsM[m_, max_] := Module[{ans = {}, rhs = {}, u, v, w, lhs, matching},
%t A385182 Do[Do[AppendTo[rhs, {v*(v + w), v, w}], {w, max}], {v, m*(m + max)}];
%t A385182 rhs = GatherBy[rhs, First];
%t A385182 Do[lhs = m*(m + u); matching = Select[rhs, #[[1, 1]] == lhs &];
%t A385182 If[Length[matching] > 0, Do[AppendTo[ans,
%t A385182 Map[{m, u, #[[2]], #[[3]]} &, matching[[1]]]], {i,
%t A385182 Length[matching]}]], {u, max}];
%t A385182 ans = Flatten[ans, 1];
%t A385182 Select[Union[Map[Sort[{#, RotateLeft[#, 2]}][[1]] &,
%t A385182 Sort[Select[DeleteDuplicates[
%t A385182 ans], {#[[1]], #[[2]]} =!= {#[[3]], #[[4]]} &]]]], #[[1]] == m &]];
%t A385182 TableForm[solns = solnsM[1, 140], TableHeadings -> {None, {"m", "u", "v", "w"}}]
%t A385182 aa = Flatten[solns]
%t A385182 Map[#[[2]] &, solns]    (* u, A385182 *)
%t A385182 Map[#[[3]] &, solns]    (* v, A385183 *)
%t A385182 Map[#[[4]] &, solns]    (* w, A385184 *)
%t A385182 (*_Peter J.C.Moses_, Jun 15 2025*)
%Y A385182 Guide to related sequences:
%Y A385182   m |    u    |    v    |    w
%Y A385182   --+---------+---------+--------
%Y A385182   1 | A385182 | A385183 | A385184
%Y A385182   2 | A385592 | A385593 | A385594
%Y A385182   3 | A385595 | A385596 | A385597
%Y A385182   4 | A385598 | A385599 | A385600
%Y A385182   --+---------+---------+--------
%Y A385182 Cf. A056924.
%K A385182 nonn
%O A385182 1,1
%A A385182 _Clark Kimberling_, Jun 23 2025