

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.


5



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
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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 irreductible 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



