A141283 -id:A141283 - cp's OEIS frontend

A084260 Triangular numbers that set a new record for number of triangular divisors.

1, 3, 6, 36, 105, 120, 210, 630, 5460, 25200, 73920, 97020, 157080, 1185030, 2031120, 2162160, 17907120, 76576500, 236215980, 7534947420, 12249176940, 78091322400, 203522319000, 666365279580, 2427046221600, 3638780505360, 12112252031880, 70233049766880, 108825865948800

Offset: 1

Jason Earls, Jun 21 2003

nonn

2*a(n) is oblong numbers that set a new record for number of oblong divisors. - Ray Chandler, Jun 29 2008

Corresponding record values for number of divisors are in A141283.

Amiram Eldar, Table of n, a(n) for n = 1..33

Cf. A000217, A076982, A076983, A141283.

Extended by Ray Chandler, Jun 21 2008

a(28)-a(29) from Amiram Eldar, Jun 23 2023

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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A084260 Triangular numbers that set a new record for number of triangular divisors.

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

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