A334185 Let m = d*q + r be the Euclidean division of m by d. The terms m of this sequence satisfy that r, q, d are consecutive positive integer terms in a geometric progression with a common integer ratio.

9, 28, 34, 65, 75, 110, 126, 132, 205, 217, 246, 258, 294, 344, 399, 436, 502, 513, 520, 579, 657, 680, 730, 810, 866, 978, 979, 1001, 1028, 1128, 1164, 1330, 1332, 1365, 1374, 1582, 1605, 1729, 1736, 1815, 1947, 2004, 2050, 2064, 2196, 2198, 2310, 2329, 2610, 2710

Offset

1,1

Comments

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

If b is the common ratio, then b is an integer >= 2.

So, when b >= 2 and r >= 1, q=r*b, d=r*b^2, then every m = r * (1+r*b^3) is a term, and the division becomes: r*(1+r*b^3) = (r*b^2) * (r*b) + r. The integers (r, r*b, r*b^2) are in geometric progression.

When (r < q < d) is solution with m = d * q + r, then, with d' = q and q' = d, m = d' * q' + r and (r < d' < q') is also a solution with another order between remainder, divisor and quotient (see last example).

m is a term if m = r * (1+r*b^3) with r >= 1 and b >= 2; so, when r = 1, A001093(n) for n > 1 are terms (see 1st example).

Links

Table of n, a(n) for n=1..50.

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

Wikipedia, Euclidean division

Example

a(2) = 28 = 9*3 + 1 with (1,3,9) and ratio = 3;
a(5) = 75 = 12*6 + 3 with (3,6,12) and ratio = 2;
a(12) = 258 = 32*8 + 2 with (2,8,32) and ratio = 4;
a(42) = 2004 = 100*20 + 4 with (r=4, q=20, d=100) but also 2004 = 20*100 + 4 with (r=4, d'=20, q'=100) both with ratio = 5:
2004 | 100             2004 |  20
     +-----                 +-----
   4 |  20                4 | 100

Mathematica

Select[Range[2000], Length @ Solve[r * (1 + r*b^3) == # && r >=1 && b >= 2, {r, b}, Integers] > 0 &] (* Amiram Eldar, Apr 18 2020 *)

Prog

(PARI) isok(m) = {for (d=1, m, if (m % d, q = m\d; r = m % d; if (!(d % q) && (d/q == q/r), return (1)); ); ); } \\ Michel Marcus, Apr 19 2020

Crossrefs

Cf. A334186 (similar, with b is an irreducible fraction).

Subsequence: A001093 \ {0, 1, 2} (for r = 1).

Sequence in context: A044999 A155473 A127629 * A267686 A024670 A141805

Adjacent sequences: A334182 A334183 A334184 * A334186 A334187 A334188

Keyword

nonn

Author

Bernard Schott, Apr 18 2020

Extensions

Name improved by Michel Marcus, Apr 19 2020

More terms from Michel Marcus, Apr 19 2020

Status

approved