A335272 For m to be a term there must exist three Euclidean divisions of m by d, d', and d", m = d*q + r = d'*q' + r' = d"*q" + r", such that (r, q, d), (r', d', q'), and (q", r", d") are three geometric progressions.

110, 132, 1332, 6162, 10712, 12210, 35156, 60762, 67340, 152490, 296480, 352242, 354620, 357006, 648830, 771762, 932190, 1197930, 2057790, 2803950, 3241800, 3310580, 4458432, 6454140, 7865220, 9613100, 10814232, 13976382, 16382256, 19267710, 53824232, 55138050

Offset

1,1

Comments

Inspired by Project Euler, Problem 141 (see link).

The terms are necessary oblong numbers >= 6.

Links

Giovanni Resta, Table of n, a(n) for n = 1..200

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

Example

For 110:
   110 | 18             110 |  6               110 | 100
       -----               ------               ---------
     2 |  6       ,       2 | 18       ,        10 |   1
For 132, see A335065.
For 1332:
  1332 | 121           1332 |  11            1332 | 1296
       ------              -------                -------
     1 |  11      ,       1 | 121      ,       36 |   1   .

Mathematica

Select[(#^2 + #) & /@ Range[2000], (n = #; AnyTrue[ Range[1 + Sqrt@ n], #^2 == Mod[n, #] Floor[n/#] &]) &] (* Giovanni Resta, Jun 03 2020 *)

Prog

(PARI) isob(n) = my(m=sqrtint(n)); m*(m+1)==n; \\ A002378
isgd(n) = {for(d=1, n, if((n\d)*(n%d)==d^2, return(1))); return(0)}; \\ A127629
isok(n) = isob(n) && isgd(n); \\ Michel Marcus, May 30 2020

Crossrefs

Intersection of A127629 and A002378.

Cf. A334185, A334186, A335064, A335065.

Sequence in context: A039447 A345386 A095611 * A358255 A307534 A200070

Adjacent sequences: A335269 A335270 A335271 * A335273 A335274 A335275

Keyword

nonn

Author

Bernard Schott, May 30 2020

Extensions

More terms from Michel Marcus, May 30 2020

a(18)-a(26) from Jinyuan Wang, May 30 2020

Terms a(27) and beyond from Giovanni Resta, Jun 03 2020

Status

approved