A335065 Let m = d*q + r be the Euclidean division of m by d. The terms m of this sequence satisfy that d, q, r are consecutive positive integer terms in a geometric progression but not necessarily in that order.

6, 9, 12, 20, 28, 30, 34, 42, 56, 58, 65, 72, 75, 90, 110, 126, 132, 156, 182, 201, 205, 210, 217, 224, 240, 246, 254, 258, 272, 294, 306, 342, 344, 380, 384, 399, 420, 436, 462, 498, 502, 506, 513, 516, 520, 552, 579, 600, 650, 657, 680, 690, 702, 730, 756, 786

Offset: 1

Author

Bernard Schott, May 23 2020

Keywords

• nonn

Comments

Inspired by the problem 141 of Project Euler (see link).

There exist 3 possibilities to get such terms m that satisfy that d, q, r are consecutive positive integer terms in a geometric progression but not necessarily in that order:

-> the geometric progression is r < q < d (A127629).

-> the geometric progression is r < d < q (same terms of A127629).

-> the geometric progression is q < r < d (A002378 \ {0,2} = oblong numbers >= 6).

Some numbers have a geometric progression solution in the 3 cases (132, 1332, 6162, ...) [see examples].

Examples

			Examples with r < q < d, r < d < q, q < r <d:
   34 | 8        75 |  6           42 | 12
      ----          -----             -----
    2 | 4     ,   3 | 12     ,      6 |  3
The 3 possible divisions by 132:
  132 | 16      132 |  8          132 | 121
      -----         ------            ------
    4 |  8   ,    4 | 16     ,     11 |   1.
		

Links

• Project Euler, Problem 141: Investigating progressive numbers, n, which are also square

Crossrefs

Equals A127629 Union A002378 \ {0,2}.

Subsequences: A334185, A334186, A335064.

Programs

Mathematica

mx = 800; Union@ Reap[ Do[y = x+1; While[(z = y^2/x) < mx, If[ IntegerQ@ z, If[(m = z y + x) <= mx, Sow@ m]; If[(m = z x + y) <= mx, Sow@ m]]; y++], {x, mx}]][[2, 1]] (* Giovanni Resta, May 24 2020 *)

PARI

isok(n) = {my(r, d); for (q=2, n-1, if (r=(n % q), d = n\q; if ((r*d == q^2) || (r*q == d^2) || (q*d == r^2), return (1));););} \\ Michel Marcus, May 25 2020