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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.
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%I #35 Jun 04 2020 07:58:35

%S 110,132,1332,6162,10712,12210,35156,60762,67340,152490,296480,352242,

%T 354620,357006,648830,771762,932190,1197930,2057790,2803950,3241800,

%U 3310580,4458432,6454140,7865220,9613100,10814232,13976382,16382256,19267710,53824232,55138050

%N 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.

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

%C The terms are necessary oblong numbers >= 6.

%H Giovanni Resta, <a href="/A335272/b335272.txt">Table of n, a(n) for n = 1..200</a>

%H Project Euler, <a href="https://projecteuler.net/problem=141">Problem 141: Investigating progressive numbers, n, which are also square</a>

%e For 110:

%e 110 | 18 110 | 6 110 | 100

%e ----- ------ ---------

%e 2 | 6 , 2 | 18 , 10 | 1

%e For 132, see A335065.

%e For 1332:

%e 1332 | 121 1332 | 11 1332 | 1296

%e ------ ------- -------

%e 1 | 11 , 1 | 121 , 36 | 1 .

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

%o (PARI) isob(n) = my(m=sqrtint(n)); m*(m+1)==n; \\ A002378

%o isgd(n) = {for(d=1, n, if((n\d)*(n%d)==d^2, return(1))); return(0)}; \\ A127629

%o isok(n) = isob(n) && isgd(n); \\ _Michel Marcus_, May 30 2020

%Y Intersection of A127629 and A002378.

%Y Cf. A334185, A334186, A335064, A335065.

%K nonn

%O 1,1

%A _Bernard Schott_, May 30 2020

%E More terms from _Michel Marcus_, May 30 2020

%E a(18)-a(26) from _Jinyuan Wang_, May 30 2020

%E Terms a(27) and beyond from _Giovanni Resta_, Jun 03 2020