A385476 - OEIS

A385476

Values of u in the quartets (1, u, v, w) of type 3; i.e., values of u for solutions to (1 - u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

5, 7, 7, 9, 9, 10, 11, 11, 13, 13, 13, 13, 15, 15, 16, 16, 17, 17, 17, 19, 19, 19, 19, 21, 21, 21, 21, 22, 22, 23, 23, 25, 25, 25, 25, 25, 25, 26, 27, 27, 28, 28, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 33, 33, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 37, 37

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OFFSET

1,1

COMMENTS

A 4-tuple (m, u, v, w) is a quartet of type 3 if m, u, v, w are distinct positive integers such that m < v and m*(m - u) = v*(v - w). Here, the values of u are arranged 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.

LINKS

Table of n, a(n) for n=1..66.

EXAMPLE

First 20 quartets (1,u,v,w) of type 3:

m u v w

1 5 2 4

1 7 2 5

1 7 3 5

1 9 2 6

1 9 4 6

1 10 3 6

1 11 2 7

1 11 5 7

1 13 2 8

1 13 3 7

1 13 4 7

1 13 6 8

1 15 2 9

1 15 7 9

1 16 3 5

1 16 3 8

1 17 2 10

1 17 4 8

1 17 8 10

1 19 2 11

1(1-11) = 5(5-7), so (1, 11, 5, 7) is in the list.

MATHEMATICA

solnsM[m_Integer?Positive, u_Integer?Positive] :=

Module[{n = m (m - u), nn, sgn, ds, tups}, If[n == 0, Return[{}]];

sgn = Sign[n]; nn = Abs[n];

ds = Divisors[nn];

If[sgn > 0, ds = Select[ds, # < nn/# &]];

tups = ({m, u, nn/#, nn/# - sgn #} & /@ ds);

Select[tups, #[[3]] > 1 && #[[4]] > 0 && #[[2]] =!= #[[4]](*&&

Length@DeleteDuplicates[#]==4*)&]];

(solns =

Sort[Flatten[Map[solnsM[1, #] &, Range[2, 30]], 1]]) // ColumnForm

Map[#[[2]] &, solns] (*A385476*)

Map[#[[3]] &, solns] (*A163870*)

Map[#[[4]] &, solns] (*A385246*)

(* Peter J. C. Moses, Aug 22 2025 *)

CROSSREFS

Cf. A385182 (type 1), A386218 (type 2), A386631, A385246.

Sequence in context: A086056 A154475 A375336 * A264527 A137755 A258316

Adjacent sequences: A385473 A385474 A385475 * A385477 A385478 A385479

KEYWORD

nonn

AUTHOR

Clark Kimberling, Aug 16 2025

STATUS

approved