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A085780 Numbers that are a product of 2 triangular numbers.

%I A085780 #56 Jul 11 2024 14:35:21
%S A085780 0,1,3,6,9,10,15,18,21,28,30,36,45,55,60,63,66,78,84,90,91,100,105,
%T A085780 108,120,126,135,136,150,153,165,168,171,190,198,210,216,225,231,234,
%U A085780 253,270,273,276,280,300,315,325,330,351,360,378,396,406,408,420,435,441
%N A085780 Numbers that are a product of 2 triangular numbers.
%C A085780 Is there a fast algorithm for detecting these numbers? - _Charles R Greathouse IV_, Jan 26 2013
%C A085780 The number of rectangles with positive width 1<=w<=i and positive height 1<=h<=j contained in an i*j rectangle is t(i)*t(j), where t(k)=A000217(k), see A096948. - _Dimitri Boscainos_, Aug 27 2015
%H A085780 T. D. Noe, <a href="/A085780/b085780.txt">Table of n, a(n) for n = 1..10000</a>
%F A085780 Conjecture: There are about sqrt(x)*log(x) terms up to x. - _Charles R Greathouse IV_, Jul 11 2024
%e A085780 18 = 3*6 = t(2)*t(3) is a product of two triangular numbers and therefore in the sequence.
%p A085780 isA085780 := proc(n)
%p A085780      local d;
%p A085780      for d in numtheory[divisors](n) do
%p A085780         if d^2 > n then
%p A085780             return false;
%p A085780         end if;
%p A085780         if isA000217(d) then
%p A085780             if isA000217(n/d) then
%p A085780                 return true;
%p A085780             end if;
%p A085780         end if;
%p A085780     end do:
%p A085780     return false;
%p A085780 end proc:
%p A085780 for n from 1 to 1000 do
%p A085780     if isA085780(n) then
%p A085780         printf("%d,",n) ;
%p A085780     end if ;
%p A085780 end do: # _R. J. Mathar_, Nov 29 2015
%t A085780 t1 = Table[n (n+1)/2, {n, 0, 100}];Select[Union[Flatten[Outer[Times, t1, t1]]], # <= t1[[-1]] &] (* _T. D. Noe_, Jun 04 2012 *)
%o A085780 (PARI) A003056(n)=(sqrtint(8*n+1)-1)\2
%o A085780 list(lim)=my(v=List([0]),t); for(a=1, A003056(lim\1), t=a*(a+1)/2; for(b=a, A003056(lim\t), listput(v,t*b*(b+1)/2))); vecsort(Vec(v),,8) \\ _Charles R Greathouse IV_, Jan 26 2013
%o A085780 (Python)
%o A085780 from itertools import count, islice
%o A085780 from sympy import divisors, integer_nthroot
%o A085780 def A085780_gen(startvalue=0): # generator of terms
%o A085780     if startvalue <= 0:
%o A085780         yield 0
%o A085780     for n in count(max(startvalue,1)):
%o A085780         for d in divisors(m:=n<<2):
%o A085780             if d**2 > m:
%o A085780                 break
%o A085780             if integer_nthroot((d<<2)+1,2)[1] and integer_nthroot((m//d<<2)+1,2)[1]:
%o A085780                 yield n
%o A085780                 break
%o A085780 A085780_list = list(islice(A085780_gen(),10)) # _Chai Wah Wu_, Aug 28 2022
%Y A085780 Cf. A000217, A085782, A068143, A000537 (subsequence), A006011 (subsequence), A033487 (subsequence), A188630 (subsequence).
%Y A085780 Cf. A072389 (this times 4).
%K A085780 nonn
%O A085780 1,3
%A A085780 _Jon Perry_, Jul 23 2003
%E A085780 More terms from _Max Alekseyev_ and _Jon E. Schoenfield_, Sep 04 2009