cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Views

Author

Robert G. Wilson v, Jan 09 2024

Keywords

Examples

			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.

Links

Crossrefs

Cf. A000217, A002378, A076982, A076983.
Essentially the same as A225400.

Programs

  • Mathematica
    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

Formula

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

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

Views

Author

Amarnath Murthy, Oct 25 2002

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A000217, A076982, A084260.

Extensions

More terms from Lior Manor, Nov 06 2002
More terms from Ray Chandler, Jun 29 2008

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

Views

Author

Jason Earls, Jun 21 2003

Keywords

Comments

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.

Links

Crossrefs

Cf. A000217, A076982, A076983, A141283.

Extensions

Extended by Ray Chandler, Jun 21 2008
a(28)-a(29) from Amiram Eldar, Jun 23 2023

A137281 Numbers k such that T(k) is not divisible by T(i), 1 < i < k, where T(k) = k-th triangular number A000217(k).

Original entry on oeis.org

2, 4, 7, 10, 13, 16, 22, 25, 28, 31, 34, 37, 43, 46, 49, 52, 58, 61, 67, 70, 73, 76, 82, 85, 88, 94, 97, 103, 106, 118, 121, 127, 130, 133, 136, 142, 145, 148, 151, 157, 163, 166, 169, 172, 178, 187, 190, 193, 196, 202, 205, 208, 211, 214, 217, 226, 229, 232, 238, 241
Offset: 1

Views

Author

Zak Seidov, Mar 14 2008

Keywords

Comments

All terms > 5 in A005383 are here. - Zak Seidov, Jun 20 2013
All terms except 2 are congruent to 1 (mod 3). This is required for 3 not to be a divisor of T(n). - Franklin T. Adams-Watters, Dec 10 2019
Conjecture: a(n) ~ C * n * log(n) for some constant C, in analogy with the prime number theorem (see A000040). - Harry Richman, Mar 05 2025

Examples

			T(4)=10 is not divisible by lesser T's 3, 6;
T(7)=28 is not divisible by lesser T's 3, 6, 10, 15, 21.

Links

Crossrefs

Cf. A000217, A005383, A076982, A226863.

Programs

  • Mathematica
    nn = 241; tri = Table[n*(n+1)/2, {n, nn}]; Select[Range[2, nn], ! MemberQ[Mod[tri[[#]], Take[tri, {2, # - 1}]], 0] &] (* T. D. Noe, Apr 12 2011 *)

Formula

n such that A076982(n) = 2. - T. D. Noe, Apr 12 2011
A000217(a(n)) = A226863(n). - Zak Seidov, Jun 20 2013

A141283 Record values for number of divisors from A084260.

Original entry on oeis.org

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

Views

Author

Ray Chandler, Jun 21 2008

Keywords

Crossrefs

Cf. A076982, A084260.

Formula

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

Extensions

a(28)-a(33) 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

Views

Author

Alex Ratushnyak, May 06 2013

Keywords

Comments

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

Examples

			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.

Crossrefs

Cf. A000217, A225400.
Cf. A076982, A076983, A084260, A137281, A141283.

Programs

  • C
    #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;
}
  • Maple
    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
  • Mathematica
    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 *)

Formula

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

A350682 Möbius values of triangular numbers under divisibility relation.

Original entry on oeis.org

1, -1, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 1, -1, 0, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0, -1, 0, 1, 0, 0, 0, -1, 2, 0, -1, 0, 1, -1, 0, 0, -1, 0, 1, 2, 1, 0, -1, 1, 1, -1, 0, 1, 0, 1, 0, -1, 0, 0, -1, 0, 0, -1, 0, 1, -1, 1, 0, 0, 0, 0, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0
Offset: 1

Views

Author

Rohan Pandey, Harry Richman, Jan 11 2022

Keywords

Comments

Consider the partial order whose elements are the triangular numbers (T(n) (A000217)) and whose order relation is integer divisibility. Then a(n) is the value mu(T(1), T(n)) of the Möbius function of this partial order.

Links

Crossrefs

Cf. A076982, A008683, A000217.

Programs

  • Mathematica
    ZetaM = Table[If[Mod[i*(i + 1), j*(j + 1)] == 0, 1, 0], {i, 100}, {j, 100}];
MobiusM = LinearSolve[ZetaM, UnitVector[100, 1]] (* Harry Richman, Jan 23 2022 *)
  • PARI
    lista(nn) = {my(v=vector(nn, k, k*(k+1)/2)); my(m=matrix(nn, nn, n, k, ! (v[n] % v[k]))); m = 1/m; vector(nn, k, m[k, 1]);} \\ Michel Marcus, Jan 19 2022
  • Python
    from sympy import *
triangular_numbers = ([(x * (x + 1) // 2) for x in range(1, 101)])
def Mobius_Matrix(lst):
    zeta_array = [[0 if n % m != 0 else 1 for n in lst] for m in lst]
    return Matrix(zeta_array) ** -1
M = Mobius_Matrix(triangular_numbers)
N = M[0, :].tolist()
print(N[0])
Showing 1-7 of 7 results.

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