A188630 -id:A188630 - cp's OEIS frontend

A212614 Least k > 1 such that the product tri(n) * tri(k) is triangular, or zero if no such k exists, where tri(k) is the k-th triangular number.

2, 5, 3, 6, 2, 4, 10, 0, 13, 7, 5, 4, 9, 3, 20, 208, 185, 14, 5, 2, 6, 14, 12, 115, 55, 37, 748, 11, 12, 1358, 90, 90, 6, 3, 21, 11, 26, 10, 33, 21, 265, 51, 61, 75, 96, 131, 201, 411, 0, 10, 7, 148, 113, 92, 4, 68, 364, 329, 50, 5083, 43, 329594, 38, 36, 2414

Offset: 1

T. D. Noe, May 31 2012

nonn

That is, tri(k) = k(k+1)/2. It is provable that a(8) and a(49) are zero.

Other terms that are zero are given in sequence A001108. Note that a(71) = 2076978. In general, a Pell equation of the form x^2 = 1 + 2*n(n+1)*y*(y+1) must be solved to find a(n). - T. D. Noe, Jun 03 2012

			For n = 2, tri(n) = 3 and the first k is 5 because tri(5) = 15 and 3*15 = 45 is triangular.

Cf. A188630 (triangular numbers that are tri(x) * tri(y) for some x,y > 1).
Cf. A212615 (similar sequence for pentagonal numbers).
Cf. A000217 (triangular numbers).

kMax = 10^6; TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; Table[t = n*(n+1)/2; k = 2; While[t2 = k*(k+1)/2; k < kMax && ! TriangularQ[t*t2], k++]; If[k == kMax, 0, k], {n, 65}]

A225440 Triangular numbers that are the product of three distinct triangular numbers greater than 1.

Original entry on oeis.org

378, 630, 990, 3240, 4095, 4950, 5460, 9180, 15400, 19110, 25200, 31878, 37128, 37950, 39060, 52650, 61425, 79800, 97020, 103740, 105570, 122265, 145530, 157080, 161028, 176715, 192510, 221445, 265356, 288420, 304590, 306936, 346528, 437580, 500500, 545490, 583740

Offset: 1

Alex Ratushnyak, May 08 2013

nonn

Triangular numbers of the form triangular(x) * triangular(y) * triangular(z), x > y > z > 1.

			378 = 3 * 6 * 21.
630 = 3 * 10 * 21.
990 = 3 * 6 * 55.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000217, A085780, A140089, A188630, A225390.

#include 
typedef unsigned long long U64;
U64 isTriangular(U64 a) {  // ! Must be a < (1<<63)
    U64 s = sqrt(a*2);
    if (a>=(1ULL<<63)) exit(1);
    return (s*(s+1)/2 == a);
}
int compare64(const void *p1, const void *p2) {
  if (*(U64*)p1 == *(U64*)p2) return 0;
  if (*(U64*)p1 < *(U64*)p2) return -1;
  return 1;
}
#define TOP (1<<21)
U64 d[TOP];
int main() {
  U64 c, x, tx, y, ty, z, tz, p = 0;
    for (x = tx = 3; tx <= TOP; tx+=x, ++x) {
    for (y = ty = 3; ty < tx;   ty+=y, ++y) {
    for (z = tz = 3; tz < ty;   tz+=z, ++z) {
    c = tx*ty*tz;
    if (c <= TOP*18 && isTriangular(c))  d[p++] = c;
  }}}
  qsort(d, p, 8, compare64);
  for (x=c=0; cx) printf("%llu, ", y), x=y;
  return 0;
}

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

Original entry on oeis.org

36, 45, 210, 315, 360, 630, 780, 990, 1260, 1386, 1540, 1800, 2850, 2970, 3510, 3570, 3780, 4095, 4788, 4851, 6300, 7920, 8415, 8550, 8778, 9450, 11700, 11781, 14850, 15400, 15561, 16380, 17640, 17955, 18018, 18648, 19110, 20790, 21420, 21450, 21528, 25116, 25200, 26565, 26775, 26796, 27720, 28980

Offset: 1

R. J. Mathar, Nov 29 2015

nonn

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.

			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.

Chai Wah Wu, Table of n, a(n) for n = 1..10602

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

A264961ct := proc(n)
    local ct,d ;
    ct := 0 ;
    for d in numtheory[divisors](n) do
        if d^2 > n then
            return ct;
        end if;
        if isA000217(d) then
            if isA000217(n/d) then
                ct := ct+1 ;
            end if;
        end if;
    end do:
    return ct;
end proc:
for n from 1 to 30000 do
    if A264961ct(n) > 1 then
        printf("%d,",n) ;
    end if;
end do:

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 *)

from _future_ import division
mmax = 10**3
tmax, A264961_dict = mmax*(mmax+1)//2, {}
ti = 0
for i in range(1,mmax+1):
    ti += i
    p = ti*i*(i-1)//2
    for j in range(i,mmax+1):
        p += ti*j
        if p <= tmax:
            A264961_dict[p] = 2 if p in A264961_dict else 1
        else:
            break
A264961_list = sorted([i for i in A264961_dict if A264961_dict[i] > 1]) # Chai Wah Wu, Nov 29 2015

A196568 Binomial coefficients of the form C(k, 3) that are products of two other binomial coefficients of the form C(k, 3).

Original entry on oeis.org

560, 19600, 43680, 893200, 1521520, 7207200, 29269240, 2845642800, 26595476600, 249466897680

Offset: 1

Kausthub Gudipati, Oct 04 2011

a(11), if it exists, > C(5*10^6, 3). - D. S. McNeil, Oct 26 2011

			     560 = C( 16, 3) = C( 8, 3) * C( 5, 3);
   19600 = C( 50, 3) = C(16, 3) * C( 7, 3);
   43680 = C( 65, 3) = C(14, 3) * C(10, 3);
  893200 = C(176, 3) = C(30, 3) * C(12, 3).

Erich Friedman, Problem of the month October 2011

Subsequence of A364151 (more than 2 factors allowed).

Cf. A188630 (analog for triangular numbers).

Name clarified by Pontus von Brömssen, Jun 22 2023

A226389 Triangular numbers representable as triangular(x)*triangular(y)+1.

Original entry on oeis.org

1, 10, 91, 136, 946, 1711, 1891, 5671, 8911, 10585, 11026, 11935, 13861, 19306, 21736, 26335, 32131, 36856, 44551, 49141, 65341, 107416, 138601, 239086, 305371, 351541, 366796, 459361, 849556, 873181, 933661, 1100386, 1413721, 1516411, 1904176, 2297296, 2467531, 3837835

Offset: 1

Alex Ratushnyak, Jun 06 2013

nonn

Cases with x=y are included.

The associated indices in A000217 are 1, 4, 13, 16, 43, 58, 61, 106, 133, 145, 148, 154,...

			10 = 3 * 3 + 1,
91 = 6 * 15 + 1,
3837835 = 378 * 10153 + 1.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A000217, A188630, A225390.

A000217inv:=proc(n) local t1; t1:=floor(sqrt(2*n)); if n = t1*(t1+1)/2 then return t1 ; else return -1; end if; end proc:
A000217 := proc(n)
    n*(n+1)/2 ;
end proc:
isA226389 := proc(n)
    local Tx, Ty;
    if n = 1 then
        return true;
    elif A000217inv(n) >= 0 then
        for x from 0 do
            Tx := A000217(x) ;
            if Tx+1 > n then
                return false;
            end if;
            for y from 0 to x do
                Ty := A000217(y) ;
                if Tx*Ty+1 >n then
                    break;
                elif Tx*Ty+1 = n then
                    return true;
                end if;
            end do:
        end do:
    else
        false;
    end if;
end proc:
for n from 0 do
    Tn := A000217(n) ;
    if isA226389(Tn) then
        printf("%d,\n",Tn) ;
    end if;
end do: # R. J. Mathar, Jun 06 2013

A296097 Triangular numbers that can be represented as a product of two triangular numbers greater than 1, as a product of three triangular numbers greater than 1, and as a product of four triangular numbers greater than 1.

Original entry on oeis.org

25200, 61425, 145530, 500500, 749700, 828828, 1185030, 2031120, 2162160, 2821500, 4573800, 5364450, 13857480, 17907120, 20991960, 21783300, 24643710, 27162135, 29610360, 30933045, 34283340, 37191000, 40901490, 60769800, 64292130, 71216145, 76576500, 90041490

Offset: 1

Alex Ratushnyak, Dec 04 2017

nonn

Triangular numbers in the products need not be distinct, that is, duplicates in the products are allowed.

A subsequence of A188630 and of A295769.

			25200 = 210 * 120 = 120 * 21 * 10 = 28 * 15 * 10 * 6.

Cf. A000217, A188630, A295769.

A379609 Triples (x,y,z) such that T(x)*T(y) = T(z) with 2 <= x <= y, where T(n) is the n-th triangular number, sorted first by values of z, then by values of x.

Original entry on oeis.org

3, 3, 8, 2, 5, 9, 4, 6, 20, 2, 20, 35, 3, 14, 35, 4, 12, 39, 5, 11, 44, 7, 10, 55, 5, 19, 75, 3, 34, 84, 9, 13, 90, 6, 21, 98, 2, 76, 132, 6, 33, 153, 4, 55, 175, 14, 18, 189, 13, 20, 195, 12, 23, 207, 15, 20, 224, 14, 22, 230, 11, 28, 231, 12, 29, 260, 7, 51, 272, 10, 38, 285, 11, 36, 296, 5, 90, 350, 3, 143, 351, 10, 50, 374, 7, 75, 399, 9, 68, 459, 19, 34, 475

Offset: 1

Kelvin Voskuijl, Dec 27 2024

			Triples begin:
  3,  3,  8;
  2,  5,  9;
  4,  6, 20;
  2, 20, 35;
  3, 14, 35;
  4, 12, 39;
  5, 11, 44;
  7, 10, 55;
  ...
The triple 2,5,9 is in this sequence because the second triangular number (3) times the fifth (15) is the ninth (45).

Cf. A000217, A188630, A198453 (additive), A225390.

A225592 a(n) = size of the network of triangular(n). Distinct triangular numbers T and R are directly connected if R = T*k, and k>0 is a triangular number less than T. Numbers belong to the same network if there is a chain of direct connections between them.

Original entry on oeis.org

1, 1, 1, 1, 2, 7, 1, 1, 2, 8, 2, 3, 7, 7, 1, 1, 1, 5, 4, 7, 2, 3, 4, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 8, 7, 2, 4, 6, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 8, 1, 1, 1, 2, 1, 4, 1, 2, 2, 1, 1, 1, 3, 2, 3, 1, 1, 1, 1, 4, 3, 2, 5, 2, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 7, 2

Offset: 1

Alex Ratushnyak, May 11 2013

nonn

			Triangular(6) = 21 belongs to the network with the following members: 21, 105, 210, 630, 19110, 25200, 145530. So a(6) = 7.
Triangular(133) = 8911 belongs to the network with the following members: 8911, 810901, 28158760, 3104129028, 328779810450, 543633020560, 18474540054600, 45742477468287600, 1553903798634573750. So a(133) = 9.

Cf. A000217, A188630.

def isTriangular(a):
    sr = 1 << (a.bit_length() >> 1)
    a += a
    while a < sr*(sr+1):  sr>>=1
    b = sr>>1
    while b:
      s = sr+b
      if a >= s*(s+1):  sr = s
      b>>=1
    return (a==sr*(sr+1))
network = [0]*1000
readPos = writePos = 0
def addConnection(R):
  global writePos
  if isTriangular(R):
    for j in range(readPos):
      if network[j]==R:  return
    network[writePos] = R
    writePos += 1
for i in range(1, 1000000):
    readPos, writePos = 0, 1
    network[0] = i*(i+1)//2
    while readPos < writePos:
      T = network[readPos]
      readPos += 1
      n = k = 3
      while k < T:
        addConnection(T*k)
        if T % k == 0 and T//k > k:  addConnection(T//k)
        k += n
        n += 1
    print(readPos, end=', ')

A295768 Triangular numbers that can be represented as a sum of two distinct triangular numbers, and as a product of two triangular numbers greater than 1.

Original entry on oeis.org

990, 1540, 2850, 4851, 8778, 11781, 15400, 26796, 43956, 61425, 61776, 70125, 105570, 145530, 176715, 189420, 270480, 303810, 349866, 437580, 526851, 715806, 719400, 749700, 799480, 810901, 828828, 1037520, 1050525, 1185030, 1493856, 1788886, 1921780, 2001000

Offset: 1

Alex Ratushnyak, Nov 27 2017

nonn

Intersection of A188630 and A260647.

			990 is representable as a product of two triangular numbers, 990 = 660 * 15, and as a sum, 990 = 780 + 210, therefore 990 is in the sequence.

Cf. A000217, A188630, A260647.

maxTerm = 3*10^6; imax = Ceiling[(Sqrt[8*maxTerm + 1] - 1)/2];
TriangularQ[n_] := IntegerQ[Sqrt[8n + 1]];
t[op_] := Table[If[1 < i < j, op[i*(i + 1)/2 , j*(j + 1)/2], Nothing], {i, 2, imax}, {j, i + 1, imax}] // Flatten // Select[#, # <= maxTerm && TriangularQ[#]&]& // Union;
Intersection[t[Plus], t[Times]] (* Jean-François Alcover, Dec 05 2017 *)

A296098 a(n) is the smallest triangular number (A000217) that can be represented as a product of k triangular numbers greater than 1 for all k = 1,2,...,n. a(n)=-1 if no such triangular number exists.

Original entry on oeis.org

3, 36, 630, 25200, 25200, 2821500, 55030954895280, 55030954895280, 55030954895280, 55030954895280, 55030954895280, 55030954895280

Offset: 1

Alex Ratushnyak, Dec 04 2017

From Jon E. Schoenfield, Apr 21 2018: (Start)
Of the 482 triangular numbers < 55030954895280 that can be represented as a product of seven triangular numbers greater than 1, the only one that can be represented as a product of two triangular numbers greater than 1 is 218434391280, which cannot be represented as a product of 3 triangular numbers greater than 1. Thus, a(n) >= 55030954895280 for all n >= 7.
However, 55030954895280 can be represented (see Example section) as a product of k triangular numbers greater than 1 for all k in 1,2,...,12 (but not for k=13), so a(7) = a(8) = ... = a(12) = 55030954895280 (and, for each n > 12, a(n) > 55030954895280, or a(n) = -1 if no such number exists).
If, for some integer N > 12, it could be proved that a(N) = -1, then it would also be established that a(n) = -1 for every n > N. (End)

			25200 is the smallest triangular number representable as a product of 2, 3 and 4 triangular numbers, 25200 = 210 * 120 = 120 * 21 * 10 = 28 * 15 * 10 * 6.
Therefore a(4)=25200.
Also, 25200 = 28 * 10 * 10 * 3 * 3, and therefore a(5)=25200.
From _Jon E. Schoenfield_, Apr 21 2018: (Start)
Let f(k_1, k_2, ..., k_m) = Product_{j=1..m} A000217(k_j) = Product_{j=1..m} (k_j*(k_j + 1)/2). Then, since no smaller number can be represented as a product of k triangular numbers greater than 1 for all k in 1,2,...,7,
a(7) = 55030954895280
     = f(10491039)
     = f(2261, 6560)
     = f(6, 493, 6560)
     = f(28, 39, 81, 323)
     = f(17, 18, 27, 40, 116)
     = f(4, 8, 17, 28, 38, 81)
     = f(2, 3, 17, 18, 26, 28, 40)
     = f(2, 2, 2, 2, 2, 17, 144, 532)
     = f(2, 2, 2, 2, 12, 17, 18, 28, 40)
     = f(2, 2, 2, 2, 2, 2, 3, 3, 40, 2261)
     = f(2, 2, 2, 2, 2, 2, 2, 2, 16, 29, 532)
     = f(2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 40, 493)
and a(7) = a(8) = a(9) = a(10) = a(11) = a(12).
(End)

Cf. A000217, A188630, A212616, A295769, A296097.

a(n) >= A212616(n) (unless a(n) = -1). - Jon E. Schoenfield, Apr 21 2018

a(7)-a(12) from Jon E. Schoenfield, Apr 21 2018

cp's OEIS Frontend

A212614 Least k > 1 such that the product tri(n) * tri(k) is triangular, or zero if no such k exists, where tri(k) is the k-th triangular number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A225440 Triangular numbers that are the product of three distinct triangular numbers greater than 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

C

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

A196568 Binomial coefficients of the form C(k, 3) that are products of two other binomial coefficients of the form C(k, 3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A226389 Triangular numbers representable as triangular(x)*triangular(y)+1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A296097 Triangular numbers that can be represented as a product of two triangular numbers greater than 1, as a product of three triangular numbers greater than 1, and as a product of four triangular numbers greater than 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

A379609 Triples (x,y,z) such that T(x)*T(y) = T(z) with 2 <= x <= y, where T(n) is the n-th triangular number, sorted first by values of z, then by values of x.

Views

Author

Keywords

Examples

Crossrefs

A225592 a(n) = size of the network of triangular(n). Distinct triangular numbers T and R are directly connected if R = T*k, and k>0 is a triangular number less than T. Numbers belong to the same network if there is a chain of direct connections between them.

Views

Author

Keywords

Examples

Crossrefs

Programs

Python

A295768 Triangular numbers that can be represented as a sum of two distinct triangular numbers, and as a product of two triangular numbers greater than 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs