A084260 -id:A084260 - cp's OEIS frontend

A141283 Record values for number of divisors from A084260.

1, 2, 3, 4, 5, 6, 8, 10, 13, 15, 16, 19, 20, 23, 24, 32, 37, 42, 51, 52, 59, 66, 74, 79, 88, 89, 94, 98, 129, 133, 154, 172, 228

Offset: 1

Ray Chandler, Jun 21 2008

Cf. A076982, A084260.

a(n) = A076982(A084260(n)).

a(28)-a(33) from Amiram Eldar, Jun 23 2023

A076983 Smallest triangular number divisible by exactly n triangular numbers.

Original entry on oeis.org

1, 3, 6, 36, 105, 120, 780, 210, 990, 630, 19110, 7140, 5460, 176715, 25200, 73920, 103740, 145530, 97020, 157080, 4744740, 2386020, 1185030, 2031120, 3483480, 13857480, 8817900, 118572300, 88438350, 39209940, 286071240, 2162160, 472612140, 953993040, 668883600

Offset: 1

Amarnath Murthy, Oct 25 2002

nonn

2*a(n) = smallest oblong number with exactly n oblong divisors. - Ray Chandler, Jun 29 2008

			a(5) = 105 has 5 divisors which are triangular numbers (1, 3, 15, 21 and 105).

Amiram Eldar, Table of n, a(n) for n = 1..155 (terms 1..80 from Ray Chandler)

Cf. A000217, A076982, A084260.

More terms from Lior Manor, Nov 06 2002

More terms from Ray Chandler, Jun 29 2008

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

Original entry on oeis.org

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

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.

A319788 Tetrahedral numbers divisible by a record number of smaller tetrahedral numbers.

Original entry on oeis.org

1, 4, 20, 120, 560, 19600, 27720, 1521520, 7207200, 2845642800, 4170866700, 249466897680, 9117204216120, 1723262134513920, 2525472914524560, 189169152233901840, 1782424363173854400, 28708458878287188000, 15137401000857582807360, 32632841312905676442600, 647550654467707884653760

Offset: 1

Torlach Rush, Sep 27 2018

nonn

For known terms > 1:
- a(n) is divisible by a square.
- a(n) is divisible by 20, n > 2.
The record numbers of tetrahedral divisors corresponding to terms a(1)-a(21) are 0, 1, 3, 4, 6, 7, 11, 16, 20, 23, 25, 32, 39, 44, 53, 57, 58, 64, 69, 72, 84.
a(18)..a(21) are divisible by 25878772920. If a(22)..a(28) are divisible by 25878772920 then they are binomial(k + 2, 3) for k in {879207614, 4118478208, 6399801198, 8921309759, 9985690350, 14992913375} having 94, 96, 98, 101, 106, 118 smaller tetrahedral divisors respectively. - David A. Corneth, Mar 22 2021

			4 is a term because it is the smallest tetrahedral number divisible by the only positive smaller tetrahedral number 1.
20 is a term because it divisible by 1,4,10, and has more divisors than each of 1,4,10, the only smaller terms.

Cf. A000292, A084260, A318556, A318701.

Subsequence of A013929.

t(n) = n*(n+1)*(n+2)/6;
f(n) = my(tn=n*(n+1)*(n+2)/6); sum(k=1, n-1, (tn % t(k)) == 0);
lista(nn) = {my(nb = - 1, new); for (n=1, nn, new = f(n); if (new > nb, print1(t(n), ", "); nb = new););} \\ Michel Marcus, Oct 02 2018

a(17)-a(18) from Giovanni Resta, Sep 28 2018

a(19)-a(21) from Bert Dobbelaere, Mar 21 2021

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A141283 Record values for number of divisors from A084260.

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A076983 Smallest triangular number divisible by exactly n triangular numbers.

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

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

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A319788 Tetrahedral numbers divisible by a record number of smaller tetrahedral numbers.

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