A076983 -id:A076983 - 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

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

Original entry on oeis.org

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

A358859 a(n) is the smallest n-gonal number divisible by exactly n n-gonal numbers.

Original entry on oeis.org

6, 36, 210, 4560, 6426, 326040, 4232250, 1969110, 296676380, 4798080, 166289760, 73765692000, 712750500, 50561280, 33944067893736, 2139168754800, 4292572951800, 1414764341760, 72461756727360, 180975331456920, 1870768457500800, 5498331930000, 153698278734000

Offset: 3

Ilya Gutkovskiy, Dec 03 2022

nonn

The corresponding indices of n-gonal numbers are 3, 6, 12, 48, 51, 330, 1100, 702, ...

			a(5) = 210, because 210 is a pentagonal number that has 5 pentagonal divisors {1, 5, 35, 70, 210} and this is the smallest such number.

Eric Weisstein's World of Mathematics, Polygonal Number
Index entries for sequences related to divisors of numbers

Cf. A005179, A076983, A358539, A358860, A358861.

a(n) = if(n<3, return()); for(k=1, oo, my(t=(k*(n*k - n - 2*k + 4))\2); if(sumdiv(t, d, ispolygonal(d, n)) == n, return(t))); \\ Daniel Suteu, Dec 04 2022

a(11)-a(25) from Daniel Suteu, Dec 04 2022

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.

A225400 First occurrence of n in A225399, or -1 if n does not appear in A225399.

Original entry on oeis.org

0, 3, 8, 14, 15, 39, 20, 44, 35, 195, 119, 104, 594, 224, 384, 455, 539, 440, 560, 3080, 2184, 1539, 2015, 2639, 5264, 4199, 15399, 13299, 8855, 23919, 2079, 30744, 43680, 36575, 14399, 5984, 58695, 113399, 47124, 107184, 12375, 78624, 98175, 73359, 111320, 242879

Offset: 0

Alex Ratushnyak, May 06 2013

nonn

Index of the least triangular number t divisible by exactly n triangular numbers bigger than 1 and less than t, or -1 if there is no such t.

Conjecture: a(n) >= 0.

Cf. A000217, A225399, A076983.

#include 
#define TOP 80
int main() {
  unsigned long long c, i, j, t, tn;
  long long f[TOP];
  memset(f, -1, sizeof(f));
  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;
        if (c

mx = 10000; tri = Table[n (n + 1)/2, {n, mx}]; nn = 20; t = Table[0, {nn}]; found = 0; n = 0; While[n < mx && found < nn, n++; cnt = 0; Do[If[Mod[tri[[n]], tri[[k]]] == 0, cnt++], {k, 2, n - 1}]; If[cnt <= nn && t[[cnt]] == 0, t[[cnt]] = n; found++]]; Join[{0}, t] (* T. D. Noe, May 07 2013 *)

A000217(a(n)) = A076983(n+1) for n>0, if the conjecture is true and a(n) >= 0.

A342808 Smallest k such that the k-th tetrahedral number is divisible by exactly n tetrahedral numbers.

Original entry on oeis.org

1, 2, 6, 4, 8, 28, 14, 48, 55, 98, 154, 54, 495, 1034, 504, 559, 208, 440, 2078, 2000, 350, 3519, 6578, 2574, 5983, 2924, 21734, 25023, 11934, 30303, 120175, 81718, 11438, 73150, 71630, 43470, 50048, 511784, 371448, 37960, 1478048, 391950, 812174, 393470, 217854, 576288

Offset: 1

David A. Corneth, Mar 22 2021

nonn

			a(6) = 28 as the 28th tetrahedral number, binomial(28+2, 3) = 4060, is divisible by 6 tetrahedral numbers namely 1, 4, 10, 20, 35 and 4060.

Cf. A000292, A076983, A318556, A319788.

first(n) = { my(res = vector(n), todo = n); for(i = 1, oo, c = f(i); if(c <= n && res[c] == 0, res[c] = i; todo--; if(todo <= 0, return(res)))) }
f(n) = my(d = divisors(binomial(n + 2, 3))); #select(x -> istetra(x), d) istetra(n)= my(k = sqrtnint(6*n, 3)); binomial(k + 2, 3) == n

A359231 a(n) is the smallest centered triangular number divisible by exactly n centered triangular numbers.

Original entry on oeis.org

1, 4, 64, 5860, 460, 74260, 14260, 1221760, 5567104, 103360, 20120860, 169096960, 1211757760, 31286787760, 31498960, 114183284260, 1553569960, 33186496960, 446613160960, 43581101074960, 274644405760, 64262632960, 121634429663260, 5786547945760

Offset: 1

Ilya Gutkovskiy, Dec 22 2022

nonn

a(25) > 10^15. a(30) = 281149511296960. - Jon E. Schoenfield, Dec 25 2022

			a(5) = 460, because 460 is a centered triangular number that has 5 centered triangular divisors {1, 4, 10, 46, 460} and this is the smallest such number.

Eric Weisstein's World of Mathematics, Centered Triangular Number
Index entries for sequences related to divisors of numbers

Cf. A005448, A076983, A300409, A358544, A358861, A359232.

// Note: the program below finds all terms through a(22) except for
//  a(20) = 43581101074960, which would be reached at k = 5390183.
a := [ 0 : n in [ 1 .. 22 ] ];
for k in [ 0 .. 550000 ] do
   c := 3*((k*(k - 1)) div 2) + 1;
   D := Divisors(c);
   n := 0;
   for d in D do
      if d mod 3 eq 1 then
         if IsSquare(((d - 1) div 3)*8 + 1) then
            n +:= 1;
         end if;
      end if;
   end for;
   if a[n] eq 0 then
      a[n] := c;
   end if;
end for;
a; // Jon E. Schoenfield, Dec 25 2022

a(8)-a(24) from Jon E. Schoenfield, Dec 25 2022

A368855 Index of the first occurrence of n in A076982.

Original entry on oeis.org

1, 2, 3, 8, 14, 15, 39, 20, 44, 35, 195, 119, 104, 594, 224, 384, 455, 539, 440, 560, 3080, 2184, 1539, 2015, 2639, 5264, 4199, 15399, 13299, 8855, 23919, 2079, 30744, 43680, 36575, 14399, 5984, 58695, 113399, 47124, 107184, 12375, 78624, 98175, 73359, 111320, 242879, 185724

Offset: 1

Robert G. Wilson v, Jan 09 2024

nonn

			a(1) = 1 since A076982(1) = 1;
a(2) = 2 since A076982(2) = 2;
a(3) = 3 since A076982(3) = 3;
a(4) = 8 since A076982(8) = 4;
a(5) = 14 since A076982(14) = 5; etc.

Robert G. Wilson v, Table of n, a(n) for n = 1..155

Cf. A000217, A002378, A076982, A076983.

Essentially the same as A225400.

k = 1; t[] := 0; f[n] := Length@ Select[ Divisors[n (n +1)/2], IntegerQ@ Sqrt[8# +1] &]; While[k < 250000, a = f@k; If[ t[a] == 0, t[a] = k]; k++]; t /@ Range@ 48

a(n) = (sqrt(8*A076983(n)+1)-1)/2. - Amiram Eldar, Jan 10 2024

cp's OEIS Frontend

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

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

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A358859 a(n) is the smallest n-gonal number divisible by exactly n n-gonal numbers.

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

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A225400 First occurrence of n in A225399, or -1 if n does not appear in A225399.

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A342808 Smallest k such that the k-th tetrahedral number is divisible by exactly n tetrahedral numbers.

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PARI

A359231 a(n) is the smallest centered triangular number divisible by exactly n centered triangular numbers.

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A368855 Index of the first occurrence of n in A076982.

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