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

1,1

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].

Links

Table of n, a(n) for n=1..56.

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

Example

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.

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 *)

Prog

(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

Crossrefs

Equals A127629 Union A002378 \ {0,2}.

Subsequences: A334185, A334186, A335064.

Sequence in context: A290130 A118782 A246356 * A315960 A106218 A315961

Adjacent sequences: A335062 A335063 A335064 * A335066 A335067 A335068

Keyword

nonn

Author

Bernard Schott, May 23 2020

Status

approved