A137281 -id:A137281 - cp's OEIS frontend

A076982 Number of triangular numbers that divide the n-th triangular number.

1, 2, 3, 2, 3, 3, 2, 4, 4, 2, 4, 4, 2, 5, 6, 2, 3, 3, 3, 8, 4, 2, 4, 6, 2, 3, 5, 2, 4, 4, 2, 5, 3, 2, 10, 4, 2, 3, 7, 3, 4, 4, 2, 9, 5, 2, 4, 6, 2, 4, 5, 2, 3, 6, 5, 6, 3, 2, 6, 6, 2, 4, 7, 3, 5, 3, 2, 4, 6, 2, 5, 5, 2, 4, 7, 2, 6, 3, 3, 9, 3, 2, 5, 10, 2, 3, 5, 2, 5, 8, 3, 4, 3, 2, 8, 4, 2, 5, 10, 3, 3, 3

Offset: 1

Amarnath Murthy, Oct 25 2002

nonn

Also number of oblong numbers that divide the n-th oblong number.

Sequence A137281 contains the indices of primitive triangular numbers; those that have no triangular divisors other than 1 and itself. - T. D. Noe, Apr 12 2011

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

Cf. A007862, A000217, A129308, A002378.
Cf. A000005, A076983, A084260.
Cf. A137281.

a[1] := 1:for i from 1 to 200 do s := 0:for j from 1 to i do if((i*(i+1)/2 mod j*(j+1)/2)=0) then s := s+1:fi:od:a[i] := s:od:seq(a[l],l=1..200);

nn = 100; tri = Table[n*(n+1)/2, {n, nn}]; Table[Count[Mod[tri[[n]], Take[tri, n]], 0], {n, nn}] (* T. D. Noe, Apr 12 2011 *)

a(n) = sumdiv(n*(n+1)/2, d, ispolygonal(d, 3)); \\ Michel Marcus, Mar 21 2023

def aupton(nn):
    tri = [i*(i+1)//2 for i in range(1, nn+1)]
    return [sum(t%t2 == 0 for t2 in tri[:j+1]) for j, t in enumerate(tri)]
print(aupton(102)) # Michael S. Branicky, Dec 10 2021

a(n) = A007862(A000217(n)) = A129308(A002378(n)). - Ray Chandler, Jun 21 2008

More terms from Lior Manor, Nov 06 2002

More terms from Sascha Kurz, Jan 26 2003

A188621 Smallest number k>1 such that k*(n-th triangular number) is also a triangular number.

Original entry on oeis.org

3, 2, 6, 12, 3, 5, 42, 56, 14, 18, 8, 10, 33, 2, 27, 240, 60, 68, 15, 3, 13, 105, 61, 67, 138, 150, 47, 51, 24, 26, 930, 117, 21, 6, 40, 66, 315, 41, 7, 231, 35, 37, 118, 5, 83, 495, 220, 230, 564, 55, 28, 147, 663, 98, 10, 50, 92, 798, 221, 229, 885, 12, 741, 615

Offset: 1

Zak Seidov, Apr 06 2011

nonn

There is a sequence of triangular numbers >3 which are not divisible by any smaller triangular number > 1, primitive triangular numbers in that sense: 3, 10, 28, 55, 91, 136, 253.... whose indices are in A137281.

(This is apparently a subsequence of A060544. - R. J. Mathar, Apr 06 2011)

			a(1)=3 because A000217(1)=1, 3*1 is triangular and k*1 for 1<k<3 is not triangular.
a(2)=2 because A000217(2)=3, 2*3 is triangular and k*3 for 1<k<2 (empty condition) is not triangular.
a(3)=6 because A000217(3)=6, 6*6 is triangular and k*6 for 1<k<6 is not triangular.
a(1000)=153 because A000217(1000)=500500, 153*500500=76576500 is triangular and k*500500 for 1<k<153 is not triangular.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A000217, A068084, A069752, A137281.

TriangularQ[n_] := IntegerQ[Sqrt[1 + 8 n]]; Table[t = (n + 1)*n/2; k = 2; While[! TriangularQ[k*t], k++]; k, {n, 100}] (* T. D. Noe, Apr 06 2011 *)
snk[n_]:=Module[{k=2},While[!OddQ[Sqrt[8k*n+1]],k++];k]; snk/@Accumulate[ Range[ 70]] (* Harvey P. Dale, Apr 29 2018 *)

a(n) = A068084(n)/A000217(n).

A226863 Triangular numbers not divisible by lesser triangular numbers > 1.

Original entry on oeis.org

3, 10, 28, 55, 91, 136, 253, 325, 406, 496, 595, 703, 946, 1081, 1225, 1378, 1711, 1891, 2278, 2485, 2701, 2926, 3403, 3655, 3916, 4465, 4753, 5356, 5671, 7021, 7381, 8128, 8515, 8911, 9316, 10153, 10585, 11026, 11476, 12403, 13366, 13861, 14365, 14878, 15931

Offset: 1

Zak Seidov, Jun 20 2013

nonn

We may coin them "prime triangulars". Certainly there are an infinity of them.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A000217, A137281 (the indices of these triangular numbers).

nn = 200; tri = Table[n (n + 1)/2, {n, nn}]; t = {}; Do[i = 2; While[i < n && Mod[tri[[n]], tri[[i]]] > 0, i++]; If[i == n, AppendTo[t, tri[[n]]]], {n, nn}]; t (* T. D. Noe, Jun 20 2013 *)

is(n)=fordiv(n, d, if(ispolygonal(d,3) && d>1 && d1 \\ Charles R Greathouse IV, Jul 29 2016

a(10000) = 1085943106 = A000217(46603).

A225399 Number of nontrivial triangular numbers dividing triangular(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 0, 2, 2, 0, 2, 2, 0, 3, 4, 0, 1, 1, 1, 6, 2, 0, 2, 4, 0, 1, 3, 0, 2, 2, 0, 3, 1, 0, 8, 2, 0, 1, 5, 1, 2, 2, 0, 7, 3, 0, 2, 4, 0, 2, 3, 0, 1, 4, 3, 4, 1, 0, 4, 4, 0, 2, 5, 1, 3, 1, 0, 2, 4, 0, 3, 3, 0, 2, 5, 0, 4, 1, 1, 7, 1, 0, 3, 8, 0, 1

Offset: 0

Alex Ratushnyak, May 06 2013

nonn

Number of triangular numbers t such that t divides triangular(n), and 1 < t < triangular(n).

			triangular(3) = 6 is divisible by triangular(2) = 3, so a(3) = 1.
triangular(8) = 36 is divisible by triangular(2) = 3 and triangular(3) = 6, so a(8) = 2.

Cf. A000217, A225400.

Cf. A076982, A076983, A084260, A137281, A141283.

#include 
int main() {
  unsigned long long c, i, j, t, tn;
  for (i = tn = 0; i < (1ULL<<32); i++) {
        for (c=0, tn += i, t = j = 3; t*2 <= tn; t+=j, ++j)
                if (tn % t == 0)  ++c;
        printf("%llu, ", c);
  }
  return 0;
}

A225399 := proc(n)
    option remember ;
    local a,tn,i;
    a := 0 ;
    tn := A000217(n) ;
    for i from 2 to n-1 do
        if modp(tn,A000217(i))=0 then
            a := a+1 ;
        end if;
    end do:
    a;
end proc:
seq(A225399(n),n=0..80) ; # R. J. Mathar, Jan 12 2024

tri = Table[n (n + 1)/2, {n, 100}]; Table[cnt = 0; Do[If[Mod[tri[[n]], tri[[k]]] == 0, cnt++], {k, 2, n - 1}]; cnt, {n, 0, Length[tri]}] (* T. D. Noe, May 07 2013 *)

a(n) = A076982(n) - 2 for n > 1.

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A076982 Number of triangular numbers that divide the n-th triangular number.

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A188621 Smallest number k>1 such that k*(n-th triangular number) is also a triangular number.

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A226863 Triangular numbers not divisible by lesser triangular numbers > 1.

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A225399 Number of nontrivial triangular numbers dividing triangular(n).

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