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%I #56 May 06 2022 13:13:51
%S 9,28,34,65,75,110,126,132,205,217,246,258,294,344,399,436,502,513,
%T 520,579,657,680,730,810,866,978,979,1001,1028,1128,1164,1330,1332,
%U 1365,1374,1582,1605,1729,1736,1815,1947,2004,2050,2064,2196,2198,2310,2329,2610,2710
%N 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.
%C Inspired by the problem 141 of Project Euler (see the link).
%C If b is the common ratio, then b is an integer >= 2.
%C 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.
%C 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).
%C 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).
%H Project Euler, <a href="https://projecteuler.net/problem=141">Problem 141: Investigating progressive numbers, n, which are also square</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Euclidean_division">Euclidean division</a>
%e a(2) = 28 = 9*3 + 1 with (1,3,9) and ratio = 3;
%e a(5) = 75 = 12*6 + 3 with (3,6,12) and ratio = 2;
%e a(12) = 258 = 32*8 + 2 with (2,8,32) and ratio = 4;
%e 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:
%e 2004 | 100 2004 | 20
%e +----- +-----
%e 4 | 20 4 | 100
%t Select[Range[2000], Length @ Solve[r * (1 + r*b^3) == # && r >=1 && b >= 2, {r, b}, Integers] > 0 &] (* _Amiram Eldar_, Apr 18 2020 *)
%o (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
%Y Cf. A334186 (similar, with b is an irreducible fraction).
%Y Subsequence: A001093 \ {0, 1, 2} (for r = 1).
%K nonn
%O 1,1
%A _Bernard Schott_, Apr 18 2020
%E Name improved by _Michel Marcus_, Apr 19 2020
%E More terms from _Michel Marcus_, Apr 19 2020
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