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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.

%I A335065 #28 Mar 26 2021 06:32:31
%S A335065 6,9,12,20,28,30,34,42,56,58,65,72,75,90,110,126,132,156,182,201,205,
%T A335065 210,217,224,240,246,254,258,272,294,306,342,344,380,384,399,420,436,
%U A335065 462,498,502,506,513,516,520,552,579,600,650,657,680,690,702,730,756,786
%N 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.
%C A335065 Inspired by the problem 141 of Project Euler (see link).
%C A335065 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:
%C A335065 -> the geometric progression is r < q < d (A127629).
%C A335065 -> the geometric progression is r < d < q (same terms of A127629).
%C A335065 -> the geometric progression is q < r < d (A002378 \ {0,2} = oblong numbers >= 6).
%C A335065 Some numbers have a geometric progression solution in the 3 cases (132, 1332, 6162, ...) [see examples].
%H A335065 Project Euler, <a href="https://projecteuler.net/problem=141">Problem 141: Investigating progressive numbers, n, which are also square</a>
%e A335065 Examples with r < q < d, r < d < q, q < r <d:
%e A335065    34 | 8        75 |  6           42 | 12
%e A335065       ----          -----             -----
%e A335065     2 | 4     ,   3 | 12     ,      6 |  3
%e A335065 The 3 possible divisions by 132:
%e A335065   132 | 16      132 |  8          132 | 121
%e A335065       -----         ------            ------
%e A335065     4 |  8   ,    4 | 16     ,     11 |   1.
%t A335065 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 *)
%o A335065 (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
%Y A335065 Equals A127629 Union A002378 \ {0,2}.
%Y A335065 Subsequences: A334185, A334186, A335064.
%K A335065 nonn
%O A335065 1,1
%A A335065 _Bernard Schott_, May 23 2020