A175497 Numbers k with the property that k^2 is a product of two distinct triangular numbers.

0, 6, 30, 35, 84, 180, 204, 210, 297, 330, 546, 840, 1170, 1189, 1224, 1710, 2310, 2940, 2970, 3036, 3230, 3900, 4914, 6090, 6930, 7134, 7140, 7245, 7440, 8976, 10710, 12654, 14175, 14820, 16296, 16380, 17220, 19866, 22770, 25172, 25944, 29103

Offset: 1

Author

Zak Seidov, May 30 2010

Keywords

• nonn

Comments

From Robert G. Wilson v, Jul 24 2010: (Start)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(End)

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

Links

• R. J. Mathar, Table of n, a(n) for n = 1..1000 replacing a b-file of _Robert G. Wilson v_ of 2010.
• R. J. Mathar, OEIS A175497, Mar 16 2023
• Project Euler, Problem 833. Square Triangle Products.
• M. Ulas, On certain diophantine equations related to triangular and tetrahedral numbers, arXiv:0811.2477 [math.NT], 2008. Theorem 5.6.

Crossrefs

From Robert G. Wilson v, Jul 24 2010: (Start)

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

Many terms are in common with A147779.

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

Cf. A152005 (two distinct tetrahedral numbers).

Programs

Maple

isA175497 := proc(n)
    local i,Ti,Tj;
    if n = 0 then
        return true;
    end if;
    for i from 1 do
        Ti := i*(i+1)/2 ;
        if Ti > n^2 then
            return false;
        else
            Tj := n^2/Ti ;
            if Tj <> Ti and type(Tj,'integer') then
                if isA000217(Tj) then  # code in A000217
                    return true;
                end if;
            end if;
        end if;
    end do:
end proc:
for n from 0 do
    if isA175497(n) then
        printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, May 26 2016

Mathematica

triangularQ[n_] := IntegerQ[Sqrt[8n + 1]];
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]]]]]];
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 *)

Python

from itertools import count, islice, takewhile
from sympy import divisors
from sympy.ntheory.primetest import is_square
def A175497_gen(startvalue=0): # generator of terms >= startvalue
    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**2A175497_list = list(islice(A175497_gen(),20)) # Chai Wah Wu, Mar 13 2023

Python

def A175497_list(n):
    def A322699_A(k, n):
        p, q, r, m = 0, k, 4*k*(k+1), 0
        while m < n:
            p, q, r = q, r, (4*k+3)*(r-q) + p
            m += 1
        return p
    def a(k, n, j):
        if n == 0: return 0
        p = A322699_A(k, n)*(A322699_A(k, n)+1)*(2*k+1) - a(k, n-1, 1)
        q = (4*k+2)*p - A322699_A(k, n)*(A322699_A(k, n)+1)//2
        m = 1
        while m < j: p, q = q, (4*k+2)*q - p; m += 1
        return p
    A = set([a(k, 1, 1) for k in range(n+1)])
    k, l, m = 1, 1, 2
    while True:
        x = a(k, l, m)
        if x < max(A):
            A |= {x}
            A  = set(sorted(A)[:n+1])
            m += 1
        else:
            if m == 1 and l == 1:
                if k > n:
                    return sorted(A)
                k += 1
            elif m > 1:
                l += 1; m = 1
            elif l > 1:
                k += 1; l, m = 1, 1
# Onur Ozkan, Mar 15 2023

Formula

a(n)^2 = A169836(n). - R. J. Mathar, Mar 12 2023