A068143 -id:A068143 - cp's OEIS frontend

A188630 Triangular numbers that are the product of two triangular numbers greater than 1.

36, 45, 210, 630, 780, 990, 1540, 2850, 3570, 4095, 4851, 8778, 11781, 15400, 17955, 19110, 21528, 25200, 26565, 26796, 33930, 37128, 40755, 43956, 61425, 61776, 70125, 79800, 105570, 113050, 122265, 145530, 176715, 189420, 192510, 246753, 270480, 303810, 349866, 437580, 500500, 526851

Offset: 1

T. D. Noe, Apr 06 2011

nonn

For squares, it is a simple matter to find squares that are the product of squares greater than 1. Is there a simple procedure for triangular numbers? That is, given n, is it easy to determine whether T(n) is the product of T(i) * T(j) for some i,j > 1?
Breiteig mentions this problem, but does not solve it. The problem can be extended to any polygonal number; for example, when is a pentagonal number the product of two pentagonal numbers? See A188660 and A188663 for the oblong and pentagonal cases.
Sequence A001571 gives the indices of triangular numbers that are 3 times another triangular number. For example, A001571(4) is 132; T(132) is 8778, which equals 3*T(76). Note that A061278 is the companion sequence, whose 4th term is 76. As with the oblong numbers covered by Breiteig, the triangular numbers in this sequence appear to satisfy linear recursions.

			210 = T(20) = 10 * 21 = T(4) * T(6).

Donovan Johnson, Table of n, a(n) for n = 1..4082
Trygve Breiteig, When is the product of two oblong numbers another oblong?, Math. Mag. 73 (2000), 120-129.

Cf. A000217 (triangular numbers), A085780 (products of two triangular numbers), A140089 (products of two triangular numbers > 1).
Subsequence of A068143 (more than 2 factors allowed).
Cf. A001571, A061278, A188660, A188663.
See also A379609.

A188630 := proc(limit) local t,E,n,k,c,b,ist; E:=NULL;
t := proc(n) option remember; iquo(n*(n+1), 2) end;
ist := proc(n) option remember; n = t(floor(sqrt(2*n))) end;
for n from 2 do
    c := t(n); if c > limit then break fi;
    for k from 2 do
        b := c*t(k); if b > limit then break fi;
        if ist(b) then E := E, b fi;
od od; sort({E}) end:
A188630(200000); # Peter Luschny, Dec 21 2017

TriangularQ[n_] := IntegerQ[Sqrt[1 + 8 n]]; TriIndex[n_] := Floor[(-1 + Sqrt[1 + 8*n])/2]; lim = 10^6; nMax = TriIndex[lim/3]; tri = Table[n (n + 1)/2, {n, 2, nMax}]; Union[Reap[Do[num = tri[[i]]*tri[[j]]; If[TriangularQ[num], Sow[num]], {i, TriIndex[Sqrt[lim]]}, {j, i, TriIndex[lim/tri[[i]]] - 1}]][[2, 1]]]
Module[{upto=530000,maxr},maxr=Ceiling[(Sqrt[1+8*Ceiling[upto/3]]-1)/2]; Union[Select[Times@@@Tuples[Rest[Accumulate[Range[maxr]]],2], IntegerQ[ Sqrt[1+8#]]&&#<=upto&]]] (* Harvey P. Dale, Jun 12 2012 *)

A085780 Numbers that are a product of 2 triangular numbers.

Original entry on oeis.org

0, 1, 3, 6, 9, 10, 15, 18, 21, 28, 30, 36, 45, 55, 60, 63, 66, 78, 84, 90, 91, 100, 105, 108, 120, 126, 135, 136, 150, 153, 165, 168, 171, 190, 198, 210, 216, 225, 231, 234, 253, 270, 273, 276, 280, 300, 315, 325, 330, 351, 360, 378, 396, 406, 408, 420, 435, 441

Offset: 1

Jon Perry, Jul 23 2003

nonn

Is there a fast algorithm for detecting these numbers? - Charles R Greathouse IV, Jan 26 2013

The number of rectangles with positive width 1<=w<=i and positive height 1<=h<=j contained in an i*j rectangle is t(i)*t(j), where t(k)=A000217(k), see A096948. - Dimitri Boscainos, Aug 27 2015

			18 = 3*6 = t(2)*t(3) is a product of two triangular numbers and therefore in the sequence.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000217, A085782, A068143, A000537 (subsequence), A006011 (subsequence), A033487 (subsequence), A188630 (subsequence).

Cf. A072389 (this times 4).

isA085780 := proc(n)
     local d;
     for d in numtheory[divisors](n) do
        if d^2 > n then
            return false;
        end if;
        if isA000217(d) then
            if isA000217(n/d) then
                return true;
            end if;
        end if;
    end do:
    return false;
end proc:
for n from 1 to 1000 do
    if isA085780(n) then
        printf("%d,",n) ;
    end if ;
end do: # R. J. Mathar, Nov 29 2015

t1 = Table[n (n+1)/2, {n, 0, 100}];Select[Union[Flatten[Outer[Times, t1, t1]]], # <= t1[[-1]] &] (* T. D. Noe, Jun 04 2012 *)

A003056(n)=(sqrtint(8*n+1)-1)\2
list(lim)=my(v=List([0]),t); for(a=1, A003056(lim\1), t=a*(a+1)/2; for(b=a, A003056(lim\t), listput(v,t*b*(b+1)/2))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Jan 26 2013

from itertools import count, islice
from sympy import divisors, integer_nthroot
def A085780_gen(startvalue=0): # generator of terms
    if startvalue <= 0:
        yield 0
    for n in count(max(startvalue,1)):
        for d in divisors(m:=n<<2):
            if d**2 > m:
                break
            if integer_nthroot((d<<2)+1,2)[1] and integer_nthroot((m//d<<2)+1,2)[1]:
                yield n
                break
A085780_list = list(islice(A085780_gen(),10)) # Chai Wah Wu, Aug 28 2022

Conjecture: There are about sqrt(x)*log(x) terms up to x. - Charles R Greathouse IV, Jul 11 2024

More terms from Max Alekseyev and Jon E. Schoenfield, Sep 04 2009

A363636 Indices of numbers of the form k^2+1, k >= 0, that can be written as a product of smaller numbers of that same form.

Original entry on oeis.org

0, 3, 7, 13, 17, 18, 21, 31, 38, 43, 47, 57, 68, 73, 91, 99, 111, 117, 123, 132, 133, 157, 183, 211, 241, 242, 253, 255, 268, 273, 293, 302, 307, 313, 322, 327, 343, 381, 413, 421, 438, 443, 463, 487, 507, 515, 553, 557, 577, 593, 601, 651, 693, 697, 703, 707

Offset: 1

Pontus von Brömssen, Jun 19 2023

nonn

For the corresponding sequence for numbers of the form k^3+1 instead of k^2+1, the only terms known to me are 0 and 26, with 26^3+1 = (2^3+1)^2*(6^3+1).

			0 is a term because 0^2+1 = 1 equals the empty product.
3 is a term because 3^2+1 = 10 = 2*5 = (1^2+1)*(2^2+1).
38 is a term because 38^2+1 = 1445 = 5*17*17 = (2^2+1)*(4^2+1)^2. (This is the first term that requires more than two factors.)

David Trimas, Table of n, a(n) for n = 1..2260

Sequences that list those terms (or their indices or some other key) of a given sequence that are products of smaller terms of the same sequence (in other words, the nonprimitive terms of the multiplicative closure of the sequence):
  A018252 (A000027),
  A034878 (A000142),
  A068143 (A000217),
  A363492 (A000041),
  A363634 (A000959),
  A363635 (A003309),
  this sequence (A002522),
  A363637 (A005563),
  A363638 (A008864),
  A363750 (A006093),
  A364151 (A000292),
  A374372 (A000326),
  A374373 (A000384),
  A374374 (A002378),
  A374375 (A007531),
  A374500 (A000330),
  A374501 (A002411),
  A374502 (A002412).

g[lst_, p_] :=
  Module[{t, i, j},
   Union[Flatten[Table[t = lst[[i]]; t[[j]] = p*t[[j]];
      Sort[t], {i, Length[lst]}, {j, Length[lst[[i]]]}], 1],
    Table[Sort[Append[lst[[i]], p]], {i, Length[lst]}]]];
multPartition[n_] :=
  Module[{i, j, p, e, lst = {{}}}, {p, e} =
    Transpose[FactorInteger[n]];
   Do[lst = g[lst, p[[i]]], {i, Length[p]}, {j, e[[i]]}]; lst];
output = Join[{0}, Flatten[Position[Table[
     test = Sqrt[multPartition[n^2 + 1][[2 ;; All]] - 1];
     Count[AllTrue[#, IntegerQ] & /@ test, True] > 0
     , {n, 707}], True]]]
(* David Trimas, Jul 23 2023 *)

A374498 Square array read by antidiagonals: row n lists n-gonal pyramidal numbers that are products of smaller n-gonal pyramidal numbers.

Original entry on oeis.org

1, 36, 1, 45, 560, 1, 210, 19600, 4900, 1, 300, 43680, 513590, 56448, 1, 378, 45760, 333833500, 127008, 4750, 1, 630, 893200, 711410700, 259200, 1926049000, 58372180608, 1

Offset: 2

Pontus von Brömssen, Jul 09 2024

If there are only finitely many solutions for a certain value of n, the rest of that row is filled with 0's.

The first term in each row is 1, because 1 is an n-gonal pyramidal number for every n and it equals the empty product.

			Array begins:
  n\k| 1     2          3            4
  ---+--------------------------------
  2  | 1    36         45          210
  3  | 1   560      19600        43680
  4  | 1  4900     513590    333833500
  5  | 1 56448     127008       259200
  6  | 1  4750 1926049000 655578709500

Cf. A080851, A374370, A374499 (second column).

Rows: A068143 (n=2 except the first term), A364151 (n=3), A374500 (n=4), A374501 (n=5), A374502 (n=6).

A364151 Tetrahedral numbers that are products of smaller tetrahedral numbers.

Original entry on oeis.org

1, 560, 19600, 43680, 45760, 893200, 1521520, 7207200, 29269240, 2845642800, 22778408800, 26595476600, 59777945920, 199910480000, 239526427140, 249466897680, 283345302240, 3280499995500, 20894643369600, 115333903584900, 408688050971200, 706949015272500, 4613394351142500

Offset: 1

Pontus von Brömssen, Jul 15 2023

nonn

			1 is a term because 1 is a tetrahedral number and equals the empty product.
560 is a term because 560 = C(16,3) = C(5,3) * C(8,3). (C(n,k) is the binomial coefficient.)
45760 is a term because 45760 = C(66,3) = C(4,3)^2 * C(5,3) * C(13,3).
3280499995500 is a term because 3280499995500 = C(27001,3) = C(4,3) * C(15,3) * C(31,3) * C(135,3).

Cf. A000292, A068143 (analog for triangular numbers), A196568 (only two factors allowed), A363636, A364152.

Row n=3 of A374498.

More terms from Jinyuan Wang, Jul 31 2023

A374370 Square array read by antidiagonals: the n-th row lists n-gonal numbers that are products of smaller n-gonal numbers.

Original entry on oeis.org

1, 4, 1, 6, 36, 1, 8, 45, 16, 1, 9, 210, 36, 10045, 1, 10, 300, 64, 11310, 2850, 1, 12, 378, 81, 20475, 61776, 6426, 1, 14, 630, 100, 52360, 79800, 9828, 1408, 1, 15, 780, 144, 197472, 103740, 35224, 61920, 265926, 1, 16, 990, 196, 230300, 145530, 60606, 67200, 391950, 69300, 1

Offset: 2

Pontus von Brömssen, Jul 07 2024

If there are only finitely many solutions for a certain value of n, the rest of that row is filled with 0's.

The first term in each row is 1, because 1 is an n-gonal number for every n and it equals the empty product.

			Array begins:
   n=2: 1,      4,       6,       8,       9,       10,       12,       14
   n=3: 1,     36,      45,     210,     300,      378,      630,      780
   n=4: 1,     16,      36,      64,      81,      100,      144,      196
   n=5: 1,  10045,   11310,   20475,   52360,   197472,   230300,   341055
   n=6: 1,   2850,   61776,   79800,  103740,   145530,   437580,   719400
   n=7: 1,   6426,    9828,   35224,   60606,  1349460,  2077992,  3333330
   n=8: 1,   1408,   61920,   67200,  276640,   297045,   870485,  1022000
   n=9: 1, 265926,  391950, 1096200, 1767546,  1787500,  9909504, 28123200
  n=10: 1,  69300, 1297890, 4257000, 5756400,  9140040,  9729720, 10648800
  n=11: 1,  79135,  792330, 2382380, 5570565, 15361500, 22230000, 49888395
  n=12: 1,   9504,   45696,  604128, 1981980,  2208465,  4798080, 13837824

Cf. A057145, A374371 (second column), A374498.

Rows: A018252 (n=2), A068143 (n=3 except first term), A062312 (n=4), A374372 (n=5), A374373 (n=6).

A374374 Oblong numbers that are products of smaller oblong numbers.

Original entry on oeis.org

12, 72, 240, 420, 600, 1056, 1260, 2352, 4032, 6480, 7140, 9120, 9900, 10920, 14280, 14520, 18360, 20592, 20880, 28392, 38220, 46872, 50400, 65280, 78120, 82656, 83232, 104652, 123552, 129960, 147840, 159600, 194040, 233772, 245520, 262656, 278256, 279312

Offset: 1

Pontus von Brömssen, Jul 07 2024

nonn

The oblong number (k^2+2*k)*(k^2+2*k+1) = A047928(k+2) is a term for all k >= 1, because it is the product of the oblong numbers k*(k+1) and (k+1)*(k+2).

			72 is a term because 72 = 8*9 = 6*12 = (2*3)*(3*4).
1056 is a term because 1056 = 32*33 = 2*2*2*132 = (1*2)*(1*2)*(1*2)*(11*12). (This is the first term that requires more than two factors.)

A188660 is a subsequence (only 2 factors allowed).

Cf. A002378, A047928, A068143, A374375.

A364152 Least n-simplex number (i.e., number of the form C(m,n) = binomial(m,n), m >= n), that can be written as a product of two or more smaller n-simplex numbers, or 0 if no such number exists.

Original entry on oeis.org

4, 36, 560, 20475, 126

Offset: 1

Pontus von Brömssen, Jul 15 2023

When n = k^2+3*k+1 is in A028387, C(n+k+3,n) = C(n+1,n) * C(n+k+1,n), so 0 != a(n) <= C(n+k+3,n). It appears that equality holds (verified for 0 <= k <= 100). In particular, a(11) = C(16,11) = 4368, a(19) = C(25,19) = 177100, a(29) = C(36,29) = 8347680, a(41) = C(49,41) = 450978066, ... .
a(34) = 4923689695575 = C(50,34) = C(35,34)*C(47,34).
a(6) > 10^29 (unless a(6) = 0). - Pontus von Brömssen, Jul 14 2024

			a(1) =     4 = C( 4, 1) = C(2,1) * C(2,1).
a(2) =    36 = C( 9, 2) = C(4,2)^2.
a(3) =   560 = C(16, 3) = C(5,3) * C(8,3). (Also, C(16,3) = C(4,3)^2 * C(7,3)).
a(4) = 20475 = C(28, 4) = C(6,4) * C(15,4).
a(5) =   126 = C( 9, 5) = C(6,5) * C(7,5).

a(1)-a(3) are the first terms greater than 1 in A018252, A068143, and A364151, respectively.

Cf. A028387.

A380825 Indices of triangular numbers that are the products of triangular numbers larger than 1.

Original entry on oeis.org

8, 9, 20, 24, 27, 35, 39, 44, 54, 55, 75, 80, 84, 90, 98, 99, 104, 132, 135, 153, 175, 189, 195, 207, 224, 230, 231, 252, 260, 272, 275, 279, 285, 296, 324, 350, 351, 374, 399, 405, 440, 455, 459, 475, 494, 539, 560, 567, 575, 594, 615, 620, 624, 665, 675

Offset: 1

Kelvin Voskuijl, Feb 04 2025

nonn

			20 is a term as the 20th triangular number is 210 = 21*10 both triangular numbers.
24 is also a term since the 24th triangular number is 300 = 3*10*10.

Cf. A000217, A003056, A068143.

a(n) = A003056(A068143(n)).

A000217(a(n)) = A068143(n).

cp's OEIS Frontend

A188630 Triangular numbers that are the product of two triangular numbers greater than 1.

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A085780 Numbers that are a product of 2 triangular numbers.

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A363636 Indices of numbers of the form k^2+1, k >= 0, that can be written as a product of smaller numbers of that same form.

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A374498 Square array read by antidiagonals: row n lists n-gonal pyramidal numbers that are products of smaller n-gonal pyramidal numbers.

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A364151 Tetrahedral numbers that are products of smaller tetrahedral numbers.

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A374370 Square array read by antidiagonals: the n-th row lists n-gonal numbers that are products of smaller n-gonal numbers.

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A374374 Oblong numbers that are products of smaller oblong numbers.

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A364152 Least n-simplex number (i.e., number of the form C(m,n) = binomial(m,n), m >= n), that can be written as a product of two or more smaller n-simplex numbers, or 0 if no such number exists.

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A380825 Indices of triangular numbers that are the products of triangular numbers larger than 1.

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