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
Keywords
Examples
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 .
Links
- Giovanni Resta, Table of n, a(n) for n = 1..200
- Project Euler, Problem 141: Investigating progressive numbers, n, which are also square
Programs
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Mathematica
Select[(#^2 + #) & /@ Range[2000], (n = #; AnyTrue[ Range[1 + Sqrt@ n], #^2 == Mod[n, #] Floor[n/#] &]) &] (* Giovanni Resta, Jun 03 2020 *)
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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
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
Comments