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A264961 Numbers that are products of two triangular numbers in more than one way.

%I A264961 #20 Jun 03 2018 02:06:29
%S A264961 36,45,210,315,360,630,780,990,1260,1386,1540,1800,2850,2970,3510,
%T A264961 3570,3780,4095,4788,4851,6300,7920,8415,8550,8778,9450,11700,11781,
%U A264961 14850,15400,15561,16380,17640,17955,18018,18648,19110,20790,21420,21450,21528,25116,25200,26565,26775,26796,27720,28980
%N A264961 Numbers that are products of two triangular numbers in more than one way.
%C A264961 One of the factors in the product may be 1 = A000217(1). We count the ways of writing n = A000217(i)*A000217(j) with i <= j, unordered factorizations.
%H A264961 Chai Wah Wu, <a href="/A264961/b264961.txt">Table of n, a(n) for n = 1..10602</a>
%e A264961 36 = 1*36 = 6*6. 45 = 1*45 = 3*15. 210 = 1*210 = 10*21. 315 = 3*105 = 15*21. 360 = 3*120 = 10*36. 630 = 1*630 = 3*210 = 6*105. 3780= 6*360 = 10 * 378 = 36*105.
%p A264961 A264961ct := proc(n)
%p A264961     local ct,d ;
%p A264961     ct := 0 ;
%p A264961     for d in numtheory[divisors](n) do
%p A264961         if d^2 > n then
%p A264961             return ct;
%p A264961         end if;
%p A264961         if isA000217(d) then
%p A264961             if isA000217(n/d) then
%p A264961                 ct := ct+1 ;
%p A264961             end if;
%p A264961         end if;
%p A264961     end do:
%p A264961     return ct;
%p A264961 end proc:
%p A264961 for n from 1 to 30000 do
%p A264961     if A264961ct(n) > 1 then
%p A264961         printf("%d,",n) ;
%p A264961     end if;
%p A264961 end do:
%t A264961 lim = 10000; t = Accumulate[Range@lim]; f[n_] := Select[{#, n/#} & /@ Select[Divisors@ n, # <= Sqrt@ n && MemberQ[t, #] &], MemberQ[t, Last@ #] &]; Select[Range@ lim, Length@ f@ # == 2 &] (* _Michael De Vlieger_, Nov 29 2015 *)
%o A264961 (Python)
%o A264961 from __future__ import division
%o A264961 mmax = 10**3
%o A264961 tmax, A264961_dict = mmax*(mmax+1)//2, {}
%o A264961 ti = 0
%o A264961 for i in range(1,mmax+1):
%o A264961     ti += i
%o A264961     p = ti*i*(i-1)//2
%o A264961     for j in range(i,mmax+1):
%o A264961         p += ti*j
%o A264961         if p <= tmax:
%o A264961             A264961_dict[p] = 2 if p in A264961_dict else 1
%o A264961         else:
%o A264961             break
%o A264961 A264961_list = sorted([i for i in A264961_dict if A264961_dict[i] > 1]) # _Chai Wah Wu_, Nov 29 2015
%Y A264961 Subsequence of A085780. A188630 and A110904 are subsequences of this.
%K A264961 nonn
%O A264961 1,1
%A A264961 _R. J. Mathar_, Nov 29 2015