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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.

%I A335272 #35 Jun 04 2020 07:58:35
%S A335272 110,132,1332,6162,10712,12210,35156,60762,67340,152490,296480,352242,
%T A335272 354620,357006,648830,771762,932190,1197930,2057790,2803950,3241800,
%U A335272 3310580,4458432,6454140,7865220,9613100,10814232,13976382,16382256,19267710,53824232,55138050
%N 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.
%C A335272 Inspired by Project Euler, Problem 141 (see link).
%C A335272 The terms are necessary oblong numbers >= 6.
%H A335272 Giovanni Resta, <a href="/A335272/b335272.txt">Table of n, a(n) for n = 1..200</a>
%H A335272 Project Euler, <a href="https://projecteuler.net/problem=141">Problem 141: Investigating progressive numbers, n, which are also square</a>
%e A335272 For 110:
%e A335272    110 | 18             110 |  6               110 | 100
%e A335272        -----               ------               ---------
%e A335272      2 |  6       ,       2 | 18       ,        10 |   1
%e A335272 For 132, see A335065.
%e A335272 For 1332:
%e A335272   1332 | 121           1332 |  11            1332 | 1296
%e A335272        ------              -------                -------
%e A335272      1 |  11      ,       1 | 121      ,       36 |   1   .
%t A335272 Select[(#^2 + #) & /@ Range[2000], (n = #; AnyTrue[ Range[1 + Sqrt@ n], #^2 == Mod[n, #] Floor[n/#] &]) &] (* _Giovanni Resta_, Jun 03 2020 *)
%o A335272 (PARI) isob(n) = my(m=sqrtint(n)); m*(m+1)==n; \\ A002378
%o A335272 isgd(n) = {for(d=1, n, if((n\d)*(n%d)==d^2, return(1))); return(0)}; \\ A127629
%o A335272 isok(n) = isob(n) && isgd(n); \\ _Michel Marcus_, May 30 2020
%Y A335272 Intersection of A127629 and A002378.
%Y A335272 Cf. A334185, A334186, A335064, A335065.
%K A335272 nonn
%O A335272 1,1
%A A335272 _Bernard Schott_, May 30 2020
%E A335272 More terms from _Michel Marcus_, May 30 2020
%E A335272 a(18)-a(26) from _Jinyuan Wang_, May 30 2020
%E A335272 Terms a(27) and beyond from _Giovanni Resta_, Jun 03 2020