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%I #16 Sep 29 2015 00:16:48
%S 300,1176,3240,7260,14196,25200,29403,41616,64980,97020,139656,195000,
%T 228150,265356,353220,461280,592416,749700,936396,1043290,1155960,
%U 1412040,1708476,2049300,2438736,2881200,3381300,3499335,3943836,4573800,5276376,6056940,6921060,7874496
%N Triangular numbers that are the product of a triangular number and a square number (both greater than 1).
%H Charles R Greathouse IV, <a href="/A253650/b253650.txt">Table of n, a(n) for n = 1..3486</a>
%e 3240 is in the sequence because 3240 is triangular number (3240=80*81/2), and 3240=10*324=(4*5/2)*(18^2), product of triangular number 10 and square number 324.
%t triQ[n_] := IntegerQ@ Sqrt[8n + 1]; lst = Sort@ Flatten@ Outer[Times, Table[ n(n + 1)/2, {n, 2, 400}], Table[ n^2, {n, 2, 200}]]; Select[ lst, triQ] (* _Robert G. Wilson v_, Jan 13 2015 *)
%o (PARI) {i=3; j=3; while(i<=10^7, k=3; p=3; c=0; while(k<i&&c==0, if(i/k==i\k&&issquare(i/k)&&i/k>1, c=k); if(c>0, print1(i, ", ")); k+=p; p+=1); i+=j; j+=1)}
%o (PARI) is(n)=if(!ispolygonal(n,3), return(0)); fordiv(core(n,1)[2], d, d>1 && ispolygonal(n/d^2,3) && n>d^2 && return(1)); 0 \\ _Charles R Greathouse IV_, Sep 29 2015
%o (PARI) list(lim)=my(v=List(),t,c); for(n=24,(sqrt(8*lim+1)-1)\2, t=n*(n+1)/2; c=core(n,1)[2]*core(n+1,1)[2]; if(valuation(t,2)\2 < valuation(c,2), c/=2); fordiv(c, d, if(d>1 && ispolygonal(t/d^2,3) && t>d^2, listput(v,t); break))); Vec(v) \\ _Charles R Greathouse IV_, Sep 29 2015
%Y Cf. A188630, A083374, A185096, A253651, A253652, A253653.
%K nonn
%O 1,1
%A _Antonio Roldán_, Jan 07 2015
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