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A265010 Numbers which are the product of two tetrahedral numbers.

%I A265010 #13 Aug 14 2024 11:24:10
%S A265010 0,1,4,10,16,20,35,40,56,80,84,100,120,140,165,200,220,224,286,336,
%T A265010 350,364,400,455,480,560,660,680,700,816,840,880,969,1120,1140,1144,
%U A265010 1200,1225,1330,1456,1540,1650,1680,1771,1820,1960,2024,2200,2240
%N A265010 Numbers which are the product of two tetrahedral numbers.
%C A265010 This is for the tetrahedral numbers A000292 what A085780 is for the triangular numbers.
%C A265010 The subsequence of numbers with more than one factorization starts 0, 560 (= 65*10 = 1*560), 19600 (= 560*15 = 19600*1), 28560 (=816 *35 = 7140*4), 43680, 292600, 416640, ...
%H A265010 Alois P. Heinz, <a href="/A265010/b265010.txt">Table of n, a(n) for n = 1..10000</a>
%F A265010 {n: n = A000292(i)*A000292(j) for some i,j>=0}.
%e A265010 Contains 480=4*120, 560=1*560, 660=4*165, 680=1*680, 700=20*35, ....
%p A265010 # reuses code of A000292
%p A265010 isA265010 := proc(n)
%p A265010     if n = 0 then
%p A265010         return true;
%p A265010     end if;
%p A265010     for d in numtheory[divisors](n) do
%p A265010         if isA000292(d) and isA000292(n/d) then
%p A265010             return true;
%p A265010         end if;
%p A265010     end do:
%p A265010     false;
%p A265010 end proc:
%p A265010 for n from 0 to 4000 do
%p A265010     if isA265010(n) then
%p A265010         printf("%d, ",n);
%p A265010     end if;
%p A265010 end do:
%t A265010 lim = 2240; t = Table[Binomial[n + 2, 3], {n, 0, 10^3}]; f[n_] := Select[{#, n/#} & /@ Select[Divisors[n], # <= Sqrt@ n && MemberQ[t, #] &], MemberQ[t, Last@ #] &]; Select[Range@ lim, Length@ f@ # > 0 &] (* _Michael De Vlieger_, Nov 30 2015 *)
%Y A265010 Cf. A085780. Subsequences: A000292, A001249, A027790, A105939, A104474, A104677.
%K A265010 nonn
%O A265010 1,3
%A A265010 _R. J. Mathar_, Nov 30 2015