A335064 Let m = d*q + r be the Euclidean division of m by d. The terms m of this sequence satisfy that q, r, d are consecutive positive integer terms in a geometric progression with a noninteger common ratio > 1.

42, 110, 156, 210, 240, 342, 420, 462, 506, 600, 702, 812, 930, 1122, 1190, 1260, 1332, 1482, 1560, 1640, 1806, 1980, 2070, 2162, 2352, 2550, 2652, 2756, 2970, 3080, 3192, 3306, 3422, 3660, 3906, 4032, 4290, 4422, 4692, 4830, 4970, 5256, 5550, 5700, 5852, 6006, 6162

Offset: 1

Author

Bernard Schott, May 22 2020

Keywords

• nonn

Comments

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

The terms of this sequence are oblong numbers m = k*(k+1) with k in A024619.

When q < r < d are consecutive terms of a geometric progression of constant b = p/s noninteger, with b>1, s>=2, p>s, it is necessary that q is a multiple of s^2, so q = q' * s^2 with q' >= 1; the Euclidean division of a term m by q becomes

p*s*q' * (1+p*s*q') = (p^2*q') * (s^2*q') + p*s*q' with k = p*s*q',

so (q, r, d) = (s^2*q', p*s*q', p^2*q') is solution. (see examples).

But, as these terms are oblong, there exists also another division where the constant ratio is the integer psq' and (q,r,d) = (1, p*s*q', (p*s*q')^2) are in geometric progression.

Examples

			Examples for 42, 110 and 156 with consecutive ratios 3/2, 5/2, 4/3:
   42 | 9         110 | 25         156 | 16
      -----           -----            -----
    6 | 4    ,     10 |  4     ,    12 |  9 ,
then with consecutive ratios 2, 10 and 12:
   42 | 12        110 | 100        156 | 144
      -----           -----            ------
    6 |  3   ,     10 |   1    ,    12 |   1.
		

Links

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

Crossrefs

Cf. A024619, A127629, A334185, A334186.

Subsequence of A002378 and of A335065.

Programs

Mathematica

Table[n*(n + 1), {n, Select[Range[80], PrimeNu[#] > 1 &]}] (* Amiram Eldar, May 23 2020 *)

PARI

apply(x->x*(x+1), select(x->!isprimepower(x), [2..80])) \\ Michel Marcus, May 23 2020

Formula

a(n) = A024619(n) * (1+A024619(n)).

a(n) = A002378(A024619(n)). - Michel Marcus, May 23 2020