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%I #9 Apr 16 2020 23:09:48
%S 0,1,3,6,10,15,18,21,28,36,45,55,60,66,78,91,105,120,136,150,153,171,
%T 180,190,210,231,253,276,300,315,325,351,378,406,435,465,496,528,561,
%U 588,595,630,666,703,741,780,820,861,900,903,946,990,1008,1035
%N Numbers that can be written as a product of one or more consecutive triangular numbers.
%H Charles R Greathouse IV, <a href="/A334129/b334129.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%t lmt = 1050; t = PolygonalNumber[3, #] & /@ Range[0, Sqrt[ 2lmt]]; f[n_] := Select[ Times @@@ Partition[t, n +1, 1], # < lmt &]; lst = {}; k = 0; While[f@k != {}, lst = Join[lst, f@k]; k++]; Union@lst (* _Robert G. Wilson v_, Apr 16 2020 *)
%o (PARI) list(lim)=if(lim<1, return(if(lim<0,[],[0]))); my(v=List([0,1]),t=1,m=2); lim\=1; while(t<=lim, listput(v,t); t=m*m++/2); for(e=1,m, for(i=3,m-e, t=factorback(Vec(v[i..i+e])); if(t>lim, break); listput(v,t))); Set(v) \\ _Charles R Greathouse IV_, Apr 16 2020
%Y Cf. A000217, A006472, A085780, A085782, A334130.
%K nonn
%O 1,3
%A _Ilya Gutkovskiy_, Apr 14 2020