A175497 - OEIS

%I #88 Jul 12 2023 11:09:05

%S 0,6,30,35,84,180,204,210,297,330,546,840,1170,1189,1224,1710,2310,

%T 2940,2970,3036,3230,3900,4914,6090,6930,7134,7140,7245,7440,8976,

%U 10710,12654,14175,14820,16296,16380,17220,19866,22770,25172,25944,29103

%N Numbers k with the property that k^2 is a product of two distinct triangular numbers.

%C From _Robert G. Wilson v_, Jul 24 2010: (Start)

%C Terms in the i-th row are products contributed with a factor A000217(i):

%C (1) 0, 6, 35, 204, 1189, 6930, 40391, 235416, 1372105, 7997214, 46611179, ...

%C (2) 30, 297, 2940, 29103, 288090, 2851797, 28229880, ...

%C (3) 84, 1170, 16296, 226974, 3161340, ...

%C (4) 180, 3230, 57960, 1040050, 18662940, ...

%C (5) 330, 7245, 159060, 3492075, 76666590, ...

%C (6) 546, 14175, 368004, 9553929, ...

%C (7) 840, 25172, 754320, 22604428, ...

%C (8) 210, 1224, 7134, 41580, 242346, 1412496, 8232630, 47983284, ...

%C (9) 1710, 64935, 2465820, 93636225, ...

%C (10) 2310, 96965, 4070220, ...

%C (11) 3036, 139590, 6418104, ...

%C (12) 3900, 194922, 9742200, ...

%C (13) 4914, 265265, 14319396, ...

%C (14) 6090, 353115, 20474580, ...

%C (15) 7440, 461160, 28584480, ...

%C (End)

%C Numbers m with property that m^2 is a product of two distinct triangular numbers T(i) and T(j) such that i and j are in the same row of the square array A(n, k) defined in A322699. - _Onur Ozkan_, Mar 17 2023

%H R. J. Mathar, <a href="/A175497/b175497.txt">Table of n, a(n) for n = 1..1000</a> replacing a b-file of _Robert G. Wilson v_ of 2010.

%H R. J. Mathar, <a href="/A175497/a175497.pdf">OEIS A175497</a>, Mar 16 2023

%H Project Euler, <a href="https://projecteuler.net/problem=833">Problem 833. Square Triangle Products</a>.

%H M. Ulas, <a href="https://arxiv.org/abs/0811.2477">On certain diophantine equations related to triangular and tetrahedral numbers</a>, arXiv:0811.2477 [math.NT], 2008. Theorem 5.6.

%F a(n)^2 = A169836(n). - _R. J. Mathar_, Mar 12 2023

%p isA175497 := proc(n)

%p     local i,Ti,Tj;

%p     if n = 0 then

%p         return true;

%p     end if;

%p     for i from 1 do

%p         Ti := i*(i+1)/2 ;

%p         if Ti > n^2 then

%p             return false;

%p         else

%p             Tj := n^2/Ti ;

%p             if Tj <> Ti and type(Tj,'integer') then

%p                 if isA000217(Tj) then  # code in A000217

%p                     return true;

%p                 end if;

%p             end if;

%p         end if;

%p     end do:

%p end proc:

%p for n from 0 do

%p     if isA175497(n) then

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

%p     end if;

%p end do: # _R. J. Mathar_, May 26 2016

%t triangularQ[n_] := IntegerQ[Sqrt[8n + 1]];

%t okQ[n_] := Module[{i, Ti, Tj}, If[n == 0, Return[True]]; For[i = 1, True, i++, Ti = i(i+1)/2; If[Ti > n^2, Return[False], Tj = n^2/Ti; If[Tj != Ti && IntegerQ[Tj], If[ triangularQ[Tj], Return[True]]]]]];

%t Reap[For[k = 0, k < 30000, k++, If[okQ[k], Print[k]; Sow[k]]]][[2, 1]] (* _Jean-François Alcover_, Jun 13 2023, after _R. J. Mathar_ *)

%o (Python)

%o from itertools import count, islice, takewhile

%o from sympy import divisors

%o from sympy.ntheory.primetest import is_square

%o def A175497_gen(startvalue=0): # generator of terms >= startvalue

%o     return filter(lambda k:not k or any(map(lambda d: is_square((d<<3)+1) and is_square((k**2//d<<3)+1), takewhile(lambda d:d**2<k,divisors(k**2)))),count(max(startvalue,0)))

%o A175497_list = list(islice(A175497_gen(),20)) # _Chai Wah Wu_, Mar 13 2023

%o (Python)

%o def A175497_list(n):

%o     def A322699_A(k, n):

%o         p, q, r, m = 0, k, 4*k*(k+1), 0

%o         while m < n:

%o             p, q, r = q, r, (4*k+3)*(r-q) + p

%o             m += 1

%o         return p

%o     def a(k, n, j):

%o         if n == 0: return 0

%o         p = A322699_A(k, n)*(A322699_A(k, n)+1)*(2*k+1) - a(k, n-1, 1)

%o         q = (4*k+2)*p - A322699_A(k, n)*(A322699_A(k, n)+1)//2

%o         m = 1

%o         while m < j: p, q = q, (4*k+2)*q - p; m += 1

%o         return p

%o     A = set([a(k, 1, 1) for k in range(n+1)])

%o     k, l, m = 1, 1, 2

%o     while True:

%o         x = a(k, l, m)

%o         if x < max(A):

%o             A |= {x}

%o             A  = set(sorted(A)[:n+1])

%o             m += 1

%o         else:

%o             if m == 1 and l == 1:

%o                 if k > n:

%o                     return sorted(A)

%o                 k += 1

%o             elif m > 1:

%o                 l += 1; m = 1

%o             elif l > 1:

%o                 k += 1; l, m = 1, 1

%o # _Onur Ozkan_, Mar 15 2023

%Y From _Robert G. Wilson v_, Jul 24 2010: (Start)

%Y A001109 (with the exception of 1), A011945, A075848 and A055112 are all proper subsets.

%Y Many terms are in common with A147779.

%Y Cf. A001108, A132596, A007654, A001108, A132593. (End)

%Y Cf. A152005 (two distinct tetrahedral numbers).

%K nonn

%O 1,2

%A _Zak Seidov_, May 30 2010