A188630 -id:A188630 - cp's OEIS frontend

A085780 Numbers that are a product of 2 triangular numbers.

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

A068143 Triangular numbers which are products of triangular numbers larger than 1.

Original entry on oeis.org

36, 45, 210, 300, 378, 630, 780, 990, 1485, 1540, 2850, 3240, 3570, 4095, 4851, 4950, 5460, 8778, 9180, 11781, 15400, 17955, 19110, 21528, 25200, 26565, 26796, 31878, 33930, 37128, 37950, 39060, 40755, 43956, 52650, 61425, 61776, 70125, 79800, 82215, 97020

Offset: 1

Amarnath Murthy, Feb 23 2002

nonn

			210 is a term as 210 = 21*10 both triangular numbers.
300 is also a term since 300 = 3*10*10.

Pontus von Brömssen, Table of n, a(n) for n = 1..3597 (all terms below 10^12).

Cf. A000217 (triangular numbers), A374374.
Except the first term, row n=3 of A374370 and row n=2 of A374498.
A188630 is a subsequence (only 2 factors allowed).

Corrected and extended by Jon E. Schoenfield, Jul 23 2006

a(38)-a(41) from Pontus von Brömssen, Jul 02 2024

A225390 Triangular numbers representable as Tx*Ty, where Tx>1 and Ty>1 are triangular numbers, in two or more ways.

Original entry on oeis.org

630, 749700, 2162160, 34283340, 76576500, 105887628, 330360660, 865924920, 2456409186, 17246794950, 35051708835, 999302826060, 3153804823260, 161708540211900, 1153195485992550, 1330786621788263640

Offset: 1

Alex Ratushnyak, May 06 2013

Triangular numbers t such that there are four triangular numbers t1, t2, t3, t4, all bigger than 1, such that t = t1 * t2 = t3 * t4.

Cf. A000217, A188630.

#include 
typedef unsigned long long U64;
U64 isTriangular(U64 a) {
    U64 sr = 1ULL<<31, s, b;
    while (a < sr*(sr+1)/2)  sr>>=1;
    for (b = sr>>1; b; b>>=1) {
        s = sr+b;
        if (a >= s*(s+1)/2)  sr = s;
    }
    return (sr*(sr+1)/2 == a);
}
int main() {
  U64 c, i, j, k, t;
  for (i = t = 0; i < (1ULL<<32); i++) {
    for (c=0, t += i, k = j = 3; k*k < t; k+=j, ++j)
      if (t%k==0 && isTriangular(t/k)) ++c;
    if (c>1) printf("%llu, ", t);
  }
  return 0;
}

a(15)-a(16) from Donovan Johnson, May 06 2013

A188663 Pentagonal numbers that are the product of two pentagonal numbers greater than 1.

Original entry on oeis.org

10045, 11310, 52360, 230300, 341055, 4048352, 6192520, 16008300, 18573282, 21430710, 44175780, 49452975, 75377337, 89579112, 174695500, 201243042, 212087876, 616116800, 755319180, 1369585525, 1557466482, 1586309340, 1625178126, 1674425676, 1744607172, 1857169860, 1868270250, 1985347551

Offset: 1

T. D. Noe, Apr 07 2011

nonn

See A188630 for the triangular case and A188660 for the oblong case.

			11310 = 5 * 2262; that is, pen(87) = pen(2) * pen(39).

Donovan Johnson, Table of n, a(n) for n = 1..361
Trygve Breiteig, When is the product of two oblong numbers another oblong?, Math. Mag. 73 (2000), 120-129.

Cf. A000326 (pentagonal numbers).

PentagonalQ[n_] := IntegerQ[(1 + Sqrt[1 + 24*n])/6]; PenIndex[n_] := Floor[(1 + Sqrt[1 + 24*n])/6]; lim = 10^10; nMax = PenIndex[lim/5]; pen = Table[n (3 n - 1)/2, {n, 2, nMax}]; Union[Reap[Do[num = pen[[i]]*pen[[j]]; If[PentagonalQ[num], Sow[num]], {i, PenIndex[Sqrt[lim]]}, {j, i, PenIndex[lim/pen[[i]]] - 1}]][[2, 1]]]

A188660 Oblong numbers that are the product of two oblong numbers.

Original entry on oeis.org

12, 72, 240, 420, 600, 1260, 2352, 4032, 6480, 7140, 9900, 14280, 14520, 20592, 28392, 38220, 46872, 50400, 65280, 78120, 83232, 104652, 123552, 129960, 159600, 194040, 233772, 279312, 291060, 331200, 390000, 456300, 485112, 530712, 609180, 613872, 699732, 706440, 809100, 852852, 922560

Offset: 1

T. D. Noe, Apr 07 2011

nonn

Breiteig writes about the finding these numbers, but does not list these numbers as a sequence. The problem can be extended to any polygonal number: for example, when is a pentagonal number the product of two pentagonal numbers? See A188630 and A188663 for the triangular and pentagonal cases.

As shown in the example, the product of consecutive oblong numbers is also oblong: oblong(n) * oblong(n+1) = oblong(n*(n+2)).

			240 = 12 * 20; that is, oblong(15) = oblong(3) * oblong(4).

Donovan Johnson, Table of n, a(n) for n = 1..5000
Trygve Breiteig, When is the product of two oblong numbers another oblong?, Math. Mag. 73 (2000), 120-129.

Cf. A002378 (oblong numbers), A188630, A188663, A374374 (more than 2 factors allowed).

OblongQ[n_] := IntegerQ[Sqrt[1 + 4 n]]; OblongIndex[n_] := Floor[(-1 + Sqrt[1 + 4*n])/2]; lim = 10^6; nMax = OblongIndex[lim/2]; obl = Table[n (n + 1), {n, nMax}]; Union[Reap[Do[num = obl[[i]]*obl[[j]]; If[OblongQ[num], Sow[num]], {i, OblongIndex[Sqrt[lim]]}, {j, i, OblongIndex[lim/obl[[i]]]}]][[2, 1]]]

A253650 Triangular numbers that are the product of a triangular number and a square number (both greater than 1).

Original entry on oeis.org

300, 1176, 3240, 7260, 14196, 25200, 29403, 41616, 64980, 97020, 139656, 195000, 228150, 265356, 353220, 461280, 592416, 749700, 936396, 1043290, 1155960, 1412040, 1708476, 2049300, 2438736, 2881200, 3381300, 3499335, 3943836, 4573800, 5276376, 6056940, 6921060, 7874496

Offset: 1

Antonio Roldán, Jan 07 2015

nonn

			3240 is in the sequence because 3240 is triangular number (3240=80*81/2), and 3240=10*324=(4*5/2)*(18^2), product of triangular number 10 and square number 324.

Charles R Greathouse IV, Table of n, a(n) for n = 1..3486

Cf. A188630, A083374, A185096, A253651, A253652, A253653.

triQ[n_] := IntegerQ@ Sqrt[8n + 1]; lst = Sort@ Flatten@ Outer[Times, Table[ n(n + 1)/2, {n, 2, 400}], Table[ n^2, {n, 2, 200}]]; Select[ lst, triQ] (* Robert G. Wilson v, Jan 13 2015 *)

{i=3; j=3; while(i<=10^7, k=3; p=3; c=0; while(k1, c=k); if(c>0, print1(i, ", ")); k+=p; p+=1); i+=j; j+=1)}

is(n)=if(!ispolygonal(n,3), return(0)); fordiv(core(n,1)[2], d, d>1 && ispolygonal(n/d^2,3) && n>d^2 && return(1)); 0 \\ Charles R Greathouse IV, Sep 29 2015

list(lim)=my(v=List(),t,c); for(n=24,(sqrt(8*lim+1)-1)\2, t=n*(n+1)/2; c=core(n,1)[2]*core(n+1,1)[2]; if(valuation(t,2)\2 < valuation(c,2), c/=2); fordiv(c, d, if(d>1 && ispolygonal(t/d^2,3) && t>d^2, listput(v,t); break))); Vec(v) \\ Charles R Greathouse IV, Sep 29 2015

A253651 Triangular numbers that are the product of a triangular number and a prime number.

Original entry on oeis.org

0, 3, 6, 15, 21, 45, 66, 78, 105, 190, 210, 231, 435, 465, 630, 861, 903, 1035, 1326, 2415, 2556, 2628, 3003, 3570, 4005, 4950, 5460, 5565, 5995, 7140, 8646, 8778, 9870, 12246, 16471, 16836, 17205, 17391, 17766, 20100, 22155, 26565, 26796, 28680, 28920, 30381, 32131, 33411, 33930, 36856

Offset: 1

Antonio Roldán, Jan 07 2015

nonn

			190 is in the sequence because it is triangular (190=19*20/2) and 190=10*19, with 10 triangular number and 19 prime number.

Harvey P. Dale, Table of n, a(n) for n = 1..259 (all terms up to and including the 10000th triangular number)

Cf. A029549 (T is 2*t), A076140 (T is 3*t), A225503 (first T to be prime(n)*t).

Cf. A188630, A253650, A253652, A253653.

N:= 10^5: # to get all terms <= N
Primes:= select(isprime, [2,seq(2*k+1,k=1..N/3)]):
select(t -> issqr(1+8*t), {seq(seq(a*(a+1)/2*p, a = 2 .. floor(sqrt(2*N/p))), p = Primes)});
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list)); # Robert Israel, Jan 07 2015

Join[{0},Module[{nn=300,trs},trs=Accumulate[Range[nn]];Select[ trs,AnyTrue[ #/trs,PrimeQ]&]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 16 2018 *)

{i=1; j=2;print1(0,", "); while(i<=10^5, k=1; p=2; c=0; while(k1, c=k); if(c>0, print1(i, ", ")); k+=p; p+=1); i+=j; j+=1)}

A253653 Triangular numbers that are the product of a square number and a prime number.

Original entry on oeis.org

3, 28, 45, 153, 171, 300, 325, 496, 2556, 2628, 3321, 4753, 4851, 7381, 8128, 13203, 19900, 25200, 25425, 29161, 29403, 56953, 64980, 65341, 101025, 166753, 195625, 209628, 320400, 354061, 388521, 389403, 468028, 662976, 664128, 749700, 750925, 780625, 781875, 936396, 1063611, 1157481

Offset: 1

Antonio Roldán, Jan 07 2015

nonn

The perfect numbers 28, 496, 8128, ... (A000396) are in the sequence, because A000396(n) = 2^(k-1)*(2^k-1) = 2^k*(2^k-1)/2 is a triangular number, and is the product of 2^(k-1) (a square when k>2) and 2^k-1 (a Mersenne prime number).
Number of terms less than 10^n: 0, 2, 7, 14, 22, 38, 68, 100, 165, 262, 420, 667, 1064, 1754, .... - Robert G. Wilson v, Jan 11 2015
This sequence is the intersection of A000217 and A229125. - Antonio Roldán, Jan 12 2015

			45 is in the sequence because it is a triangular number (45 = 9*10/2) and 45 = 9*5, with 9 a square number and 5 a prime number.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000217, A188630, A229125, A253650, A253651, A253652.

N:= 10^7: # to get all entries <= N
Tris:= {seq(x*(x+1)/2, x = 1 .. floor((sqrt(1+8*N)-1)/2))}:
Primes:= select(isprime,[2,seq(2*i+1,i=1..floor(N/8-1))]):
Tris intersect {3,seq(seq(p*y^2,y=2..floor(sqrt(N/p))),p=Primes)};
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%, list));  # Robert Israel, Jan 14 2015

tri[n_] := n(n+1)/2; fQ[n_] := Block[{exp = Sort[ Last@# & /@ FactorInteger@ n]}, exp[[1]] == 1 != exp[[2]] && Union@ Mod[ Rest@ exp, 2] == {0}]; Select[ tri@ Range@ 1500, fQ] (* Robert G. Wilson v, Jan 11 2015 *)

{i=1; j=2; while(i<=3*10^6, k=1; p=3; c=0; while(k0, print1(i, ", ")); k+=p; p+=2); i+=j; j+=1)}

lista(nn) = {for (n=1, nn, if (isprime(core(t=n*(n+1)/2)), print1(t, ", ")););} \\ Michel Marcus, Jan 12 2015

A253652 Triangular numbers that are the product of a triangular number and an oblong number.

Original entry on oeis.org

0, 6, 36, 120, 210, 300, 630, 1176, 2016, 3240, 3570, 4950, 7140, 7260, 10296, 14196, 19110, 23436, 25200, 32640, 39060, 41616, 52326, 61776, 64980, 79800, 97020, 116886, 139656, 145530, 165600, 195000, 228150, 242556, 265356, 304590, 306936, 349866, 353220, 404550, 426426, 461280

Offset: 1

Antonio Roldán, Jan 07 2015

nonn

Supersequence of A083374, because A083374(n)= n^2 * (n^2 - 1)/2 = n*(n+1)/2*n*(n-1), product of triangular number n*(n+1)/2 and oblong number n*(n-1).

			630 is in the sequence because it is a triangular number (630 = 35*36/2) and 630 = 105*6, with 105 = 14*15/2, triangular number, and 6 = 2*3, oblong number.

Cf. A002378, A000217, A188630, A083374, A253650, A253651, A253653.

{i=0;j=1;print1(0,", ");while(i<=10^6,k=1;p=2;c=0;while(k0,c=k);if(c>0,print1(i,", "));k+=p;p+=1);i+=j;j+=1)}

A295769 Triangular numbers that can be represented as a product of two triangular numbers greater than 1, and as a product of three triangular numbers greater than 1.

Original entry on oeis.org

630, 990, 4095, 15400, 19110, 25200, 37128, 61425, 79800, 105570, 122265, 145530, 176715, 192510, 437580, 500500, 749700, 828828, 1185030, 2031120, 2162160, 2821500, 4279275, 4573800, 4744740, 4959675, 5364450, 6053460, 7556328, 8817900, 13857480, 15992340

Offset: 1

Alex Ratushnyak, Nov 27 2017

nonn

Duplicates in the products are allowed.

A subsequence of A188630.

			630 = 105*6 = 21*10*3.
990 = 66*15 = 55*6*3.

Cf. A000217, A188630.

A295769 := proc(limit) local t,E,G,n,k,j,c,b,d,ist; E:=NULL; G:=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;
        for j from 2 do
            d := b*t(j); if d > limit then break fi;
            if ist(d) then G := G, d fi
od od od; {E} intersect {G} end:
A295769(200000); # Peter Luschny, Dec 21 2017

cp's OEIS Frontend

A085780 Numbers that are a product of 2 triangular numbers.

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A068143 Triangular numbers which are products of triangular numbers larger than 1.

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A225390 Triangular numbers representable as Tx*Ty, where Tx>1 and Ty>1 are triangular numbers, in two or more ways.

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A188663 Pentagonal numbers that are the product of two pentagonal numbers greater than 1.

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Mathematica

A188660 Oblong numbers that are the product of two oblong numbers.

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Mathematica

A253650 Triangular numbers that are the product of a triangular number and a square number (both greater than 1).

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A253651 Triangular numbers that are the product of a triangular number and a prime number.

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A253653 Triangular numbers that are the product of a square number and a prime number.

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