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A264961 Numbers that are products of two triangular numbers in more than one way. 2

%I

%S 36,45,210,315,360,630,780,990,1260,1386,1540,1800,2850,2970,3510,

%T 3570,3780,4095,4788,4851,6300,7920,8415,8550,8778,9450,11700,11781,

%U 14850,15400,15561,16380,17640,17955,18018,18648,19110,20790,21420,21450,21528,25116,25200,26565,26775,26796,27720,28980

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

%C 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 Chai Wah Wu, <a href="/A264961/b264961.txt">Table of n, a(n) for n = 1..10602</a>

%e 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 A264961ct := proc(n)

%p local ct,d ;

%p ct := 0 ;

%p for d in numtheory[divisors](n) do

%p if d^2 > n then

%p return ct;

%p end if;

%p if isA000217(d) then

%p if isA000217(n/d) then

%p ct := ct+1 ;

%p end if;

%p end if;

%p end do:

%p return ct;

%p end proc:

%p for n from 1 to 30000 do

%p if A264961ct(n) > 1 then

%p printf("%d,",n) ;

%p end if;

%p end do:

%t 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 (Python)

%o from __future__ import division

%o mmax = 10**3

%o tmax, A264961_dict = mmax*(mmax+1)//2, {}

%o ti = 0

%o for i in range(1,mmax+1):

%o ti += i

%o p = ti*i*(i-1)//2

%o for j in range(i,mmax+1):

%o p += ti*j

%o if p <= tmax:

%o A264961_dict[p] = 2 if p in A264961_dict else 1

%o else:

%o break

%o A264961_list = sorted([i for i in A264961_dict if A264961_dict[i] > 1]) # _Chai Wah Wu_, Nov 29 2015

%Y Subsequence of A085780. A188630 and A110904 are subsequences of this.

%K nonn

%O 1,1

%A _R. J. Mathar_, Nov 29 2015

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Last modified March 7 10:30 EST 2021. Contains 341869 sequences. (Running on oeis4.)