A309332 -id:A309332 - cp's OEIS frontend

A082183 Smallest k > 0 such that T(n) + T(k) = T(m), for some m, T(i) being the triangular numbers, n > 1.

2, 5, 9, 3, 5, 27, 10, 4, 8, 14, 17, 9, 5, 21, 135, 12, 14, 35, 6, 9, 17, 30, 12, 18, 10, 7, 54, 21, 23, 495, 42, 14, 26, 8, 49, 27, 15, 20, 98, 30, 32, 80, 9, 19, 35, 62, 45, 17, 20, 14, 99, 39, 10, 18, 54, 24, 44, 78, 81, 45, 25, 85, 153, 11, 50, 125, 20, 29, 53, 94, 97

Offset: 2

Ralf Stephan, Apr 06 2003

nonn

For 16 years this entry stood with no upper bound, and indeed with no proof that a(n) always existed. In February 2020 the following three bounds and formulas arrived. They are listed in chronological order.  Here k = k(n) denotes the smallest number such that T(n)+T(k) is a triangular number T(m) for some m = m(n). - N. J. A. Sloane, Feb 22 2020
k = T(n) - 1 is an upper bound on k(n) = a(n). For T(k) makes a huge triangle; all the elements of the T(n) triangle can be thinly plated onto the side of the big one as a single additional row, producing T(k+1) with m = k+1. - Allan C. Wechsler, Feb 19 2020
Let Q be the largest odd number < n dividing T(n). Then T(n) is the sum of Q consecutive integers, the last Q rows of the triangle T(m) with m = T(n)/Q + (Q-1)/2, giving the upper bound k <= T(n)/Q - (Q+1)/2. [This bound is now A332554, the values of Q are in A332547.] This bound is not tight: for n=9 it gives a(9) <= 6 when in fact a(9) = 4. - Michael J. Collins, Feb 19 2020
Comments from Richard C. Schroeppel, Feb 19 2020: (Start)
2T(n) = 2T(m) - 2T(k) = m^2 + m - k^2 - k = (m-k) (m + k + 1). Now (m-k) and (m+k+1) are of opposite parity. Factor 2T(n) into the product of an odd number times an even number. We can take one of these to be m-k, and the other to be m+k+1.
The factorization 2T(n) = n^2 + n gives two obvious solutions, n * (n+1) and 1 * (n^2+n). Equating these to (m-k) * (m+k+1) gives the two "trivial" solutions k=0, m=n and k=T(n)-1, m=T(n).
Unless n is a Mersenne prime, or n+1 is a Fermat prime [these are the n such that Q=1, see A068194] there will be a nontrivial odd divisor of n(n+1) other than 1, n, or n+1. Select the odd divisor d logarithmicly closest to n + 1/2 that isn't n or n+1.
Let q be the quotient n(n+1)/q. Then m-k = min(d,q) and m+k+1 = max(d,q). Solve for k, which is the required minimum k(n) = a(n).
Example:  n=5, T(n)=15, 2T(n)=30 = 3*10, d=3, q=10, k=3, m=6, 15+6 = 21. (End)

Chai Wah Wu, Table of n, a(n) for n = 2..10000
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.

Cf. A000217, A072522, values of m are in A082184, A332547.
A332554 is an upper bound on a(n).
See A055527 for a very similar sequence involving Pythagorean triples. - Bradley Klee, Feb 20 2020
See also A309332 (number of ways to write a triangular number as a sum of two triangular numbers), A309507 (... as a difference ...).

f:= proc(n) local e,t,te;
     t:= n*(n+1);
     e:= padic:-ordp(t,2);
     te:= 2^e;
     min(map(d -> (abs(te*d-t/(te*d))-1)/2, numtheory:-divisors(t/te)) minus {0}):
map(f, [$2..100]); # Robert Israel, Sep 15 2017

Table[SelectFirst[Range[10^3], Function[m, PolygonalNumber@ Floor@ Sqrt[2 m] == m][PolygonalNumber[n] + PolygonalNumber[#]] &], {n, 2, 72}] (* Michael De Vlieger, Sep 19 2017, after Maple by Robert Israel *)

for(n=2, 100, t=n*(n+1)/2; for(k=1, 10^9, u=t+k*(k+1)/2; v=floor(sqrt(2*u)); if(v*(v+1)/2==u, print1(k", "); break)))

from _future_ import division
from sympy import divisors
def A082183(n):
    t = n*(n+1)
    ds = divisors(t)
    for i in range(len(ds)//2-2,-1,-1):
        x = ds[i]
        y = t//x
        a, b = divmod(y-x,2)
        if b:
            return a
    return -1 # Chai Wah Wu, Sep 12 2017

Entry updated by N. J. A. Sloane, Feb 22 2020

A309507 Number of ways the n-th triangular number T(n) = A000217(n) can be written as the difference of two positive triangular numbers.

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 1, 2, 5, 3, 3, 3, 3, 7, 3, 1, 5, 5, 3, 7, 7, 3, 3, 5, 5, 7, 7, 3, 7, 7, 1, 3, 7, 7, 11, 5, 3, 7, 7, 3, 7, 7, 3, 11, 11, 3, 3, 5, 8, 11, 7, 3, 7, 15, 7, 7, 7, 3, 7, 7, 3, 11, 5, 3, 15, 7, 3, 7, 15, 7, 5, 5, 3, 11, 11, 7, 15, 7, 3, 9, 9, 3, 7

Offset: 1

Alois P. Heinz, Aug 05 2019

nonn

Equivalently, a(n) is the number of triples [n,k,m] with k>0 satisfying the Diophantine equation n*(n+1) + k*(k+1) - m*(m+1) = 0. Any such triple satisfies a triangle inequality, n+k > m. The n for which there is a triple [n,n,m] are listed in A053141. - Bradley Klee, Mar 01 2020; edited by N. J. A. Sloane, Mar 31 2020

			a(5) = 3: T(5) = T(6)-T(3) = T(8)-T(6) = T(15)-T(14).
a(7) = 1: T(7) = T(28)-T(27).
a(8) = 2: T(8) = T(13)-T(10) = T(36)-T(35).
a(9) = 5: T(9) = T(10)-T(4) = T(11)-T(6) = T(16)-T(13) = T(23)-T(21) = T(45)-T(44).
a(49) = 8: T(49) = T(52)-T(17) = T(61)-T(36) = T(94)-T(80) = T(127)-T(117) = T(178)-T(171) = T(247)-T(242) = T(613)-T(611) = T(1225)-T(1224).
The triples with n <= 16 are:
2, 2, 3
3, 5, 6
4, 9, 10
5, 3, 6
5, 6, 8
5, 14, 15
6, 5, 8
6, 9, 11
6, 20, 21
7, 27, 28
8, 10, 13
8, 35, 36
9, 4, 10
9, 6, 11
9, 13, 16
9, 21, 23
9, 44, 45
10, 8, 13
10, 26, 28
10, 54, 55
11, 14, 18
11, 20, 23
11, 65, 66
12, 17, 21
12, 24, 27
12, 77, 78
13, 9, 16
13, 44, 46
13, 90, 91
14, 5, 15
14, 11, 18
14, 14, 20
14, 18, 23
14, 33, 36
14, 51, 53
14, 104, 105
15, 21, 26
15, 38, 41
15, 119, 120
16, 135, 136. - _N. J. A. Sloane_, Mar 31 2020

Alois P. Heinz, Table of n, a(n) for n = 1..20000
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.

Cf. A000217, A001108, A046079 (the same for squares), A068194, A100821 (the same for primes for n>1), A309332.
See also A053141. The monotonic triples [n,k,m] with n <= k <= m are counted in A333529.
Cf. A092517, A063440.

with(numtheory): seq(tau(n*(n+1))-tau(n*(n+1)/2)-1, n=1..80); # Ridouane Oudra, Dec 08 2023

TriTriples[TNn_] := Sort[Select[{TNn, (TNn + TNn^2 - # - #^2)/(2 #),
      (TNn + TNn^2 - # + #^2)/(2 #)} & /@
    Complement[Divisors[TNn (TNn + 1)], {TNn}],
   And[And @@ (IntegerQ /@ #), And @@ (# > 0 & /@ #)] &]]
Length[TriTriples[#]] & /@ Range[100]
(* Bradley Klee, Mar 01 2020 *)

a(n) = 1 <=> n in { A068194 } \ { 1 }.
a(n) is even <=> n in { A001108 } \ { 0 }.
a(n) = number of odd divisors of n*(n+1) (or, equally, of T(n)) that are greater than 1. - N. J. A. Sloane, Apr 03 2020
a(n) = A092517(n) - A063440(n) - 1. - Ridouane Oudra, Dec 08 2023

cp's OEIS Frontend

A082183 Smallest k > 0 such that T(n) + T(k) = T(m), for some m, T(i) being the triangular numbers, n > 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Extensions