A082183 -id:A082183 - cp's OEIS frontend

A332552 a(n) = A082184(n) - A082183(n).

1, 1, 1, 3, 3, 1, 3, 6, 5, 4, 4, 7, 10, 5, 1, 9, 9, 5, 15, 14, 11, 8, 15, 13, 18, 21, 7, 15, 15, 1, 11, 22, 17, 28, 12, 19, 26, 24, 8, 21, 21, 11, 36, 30, 23, 16, 21, 35, 34, 39, 13, 27, 45, 40, 24, 38, 29, 20, 20, 31, 42, 21, 13, 55, 33, 17, 51, 46, 35

Offset: 2

N. J. A. Sloane, Feb 21 2020, at the suggestion of Allan C. Wechsler

nonn

N. J. A. Sloane, Table of n, a(n) for n = 2..10000

Cf. A082183, A082184, A332553.

A332553 a(n) = n + A082183(n) - A082184(n).

Original entry on oeis.org

1, 2, 3, 2, 3, 6, 5, 3, 5, 7, 8, 6, 4, 10, 15, 8, 9, 14, 5, 7, 11, 15, 9, 12, 8, 6, 21, 14, 15, 30, 21, 11, 17, 7, 24, 18, 12, 15, 32, 20, 21, 32, 8, 15, 23, 31, 27, 14, 16, 12, 39, 26, 9, 15, 32, 19, 29, 39, 40, 30, 20, 42, 51, 10, 33, 50, 17, 23, 35

Offset: 2

N. J. A. Sloane, Feb 21 2020, at the suggestion of Bradley Klee

nonn

Since (by definition) a(n) = n + A082183(n) - A082184(n) = - (n^2 + A082183(n)^2 - A082184(n)^2), this can be described as the distance of (n, A082183(n), A082184(n)) from a Pythagorean triple. Also a(n) > 0 for all n. See the Myers et al. link. - Bradley Klee, Feb 19 2020
Comments from N. J. A. Sloane, Feb 23 2020: (Start)
To study the lowest values taken by a(n), consider the record high values of n/a(n). The data suggests two conjectures.
Conjecture 1: The record high values of n/a(n) are j/2 + 1 for j = 2,3,4,5,... and occur at n = j*(j+1)/2 - 1.
This would imply:
Conjecture 2: Let j = 2,3,4,5,... For 1 <= n < T_j - 1, a(n) > 2*n/(j+2). (End)

N. J. A. Sloane, Table of n, a(n) for n = 2..1000
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000, April 2020.

Cf. A082183, A082184, A332552.

A055527 Shortest other leg of a Pythagorean triangle with n as length of a leg.

Original entry on oeis.org

4, 3, 12, 8, 24, 6, 12, 24, 60, 5, 84, 48, 8, 12, 144, 24, 180, 15, 20, 120, 264, 7, 60, 168, 36, 21, 420, 16, 480, 24, 44, 288, 12, 15, 684, 360, 52, 9, 840, 40, 924, 33, 24, 528, 1104, 14, 168, 120, 68, 39, 1404, 72, 48, 33, 76, 840, 1740, 11, 1860, 960, 16, 48, 72

Offset: 3

Henry Bottomley, May 22 2000

nonn

From Alex Ratushnyak, Mar 30 2014: (Start)
Least positive k such that n^2 + k^2 is a square.
For odd n, a(n) <= 4*triangular((n-1)/2), because n^2 + (4 * triangular((n-1)/2))^2 = ((n^2+1)/2) ^ 2, which is a perfect square since n is odd.
For n = 4*k+2, a(n) <= 8*triangular(k), because (4k+2)^2 + (4*k*(k+1))^2 = (4*k^2 + 4*k + 2)^2. (End)

T. D. Noe, Table of n, a(n) for n = 3..1000

Cf. A000290, A000217, A009112, A046079, A046080, A046081, A054435, A054436, A055522, A055523, A055524, A055525, A055526.

See A082183 for a similar sequence involving triangular numbers.

Table[k = 1; While[! IntegerQ[Sqrt[n^2 + k^2]], k++]; k, {n, 3, 100}] (* T. D. Noe, Apr 02 2014 *)

a(n) = sqrt(A055526(n)^2-n^2) = 2*A054436/n.

A332542 a(n) is the smallest k such that n+(n+1)+(n+2)+...+(n+k) is divisible by n+k+1.

Original entry on oeis.org

2, 7, 14, 3, 6, 47, 14, 4, 10, 20, 25, 11, 5, 31, 254, 15, 18, 55, 6, 10, 22, 44, 14, 23, 11, 7, 86, 27, 30, 959, 62, 16, 34, 8, 73, 35, 17, 24, 163, 39, 42, 127, 9, 22, 46, 92, 62, 19, 23, 15, 158, 51, 10, 20, 75, 28, 58, 116, 121, 59, 29, 127, 254, 11

Offset: 3

Scott R. Shannon, Feb 18 2020

nonn

Note that (n+(n+1)+(n+2)+...+(n+k))/(n+k+1) = A332544(n)/(n+k+1) = A082183(n-1). See the Myers et al. link for proof. - N. J. A. Sloane, Apr 30 2020

We can always take k = n^2-2*n-1, for then the sum in the definition becomes (n+1)*n*(n-1)*(n-2)/2, which is an integral multiple of n+k+1 = n*(n-1). So a(n) always exists. - N. J. A. Sloane, Feb 20 2020

			n=4: we get 4 -> 4+5=9 -> 9+6=15 -> 15+7=22 -> 22+8=30 -> 30+9=39 -> 39+10=49 -> 49+11=60, which is divisible by 12, and took k=7 steps, so a(4) = 7. Also A332543(4) = 12, A332544(4) = 60, and A082183(3) = 60/12 = 5.

Seiichi Manyama, Table of n, a(n) for n = 3..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], 2020-2021.

Cf. A332543, A332544, A082183.

See A332558-A332561 for a multiplicative analog.

grow2 := proc(n,M) local p,q,k; # searches out to a limit of M
# returns n, k (A332542(n)), n+k+1 (A332543(n)), p (A332544(n)), and q (which appears to match A082183(n-1))
for k from 1 to M do
   if ((k+1)*n + k*(k+1)/2) mod (n+k+1) = 0 then
   p := (k+1)*n+k*(k+1)/2;
   q := p/(n+k+1); return([n,k,n+k+1,p,q]);
   fi;
od:
# if no success, return -1's
[n,-1,-1,-1,-1]; end; # N. J. A. Sloane, Feb 18 2020

a[n_] := NestWhile[#1+1&,0,!IntegerQ[Divide[(#+1)*n+#*(#+1)/2,n+#+1]]&]
a/@Range[3,100] (* Bradley Klee, Apr 30 2020 *)

a(n) = my(k=1); while (sum(i=0, k, n+i) % (n+k+1), k++); k; \\ Michel Marcus, Aug 26 2021

def a(n):
    k, s = 1, 2*n+1
    while s%(n+k+1) != 0: k += 1; s += n+k
    return k
print([a(n) for n in range(3, 67)]) # Michael S. Branicky, Aug 26 2021

def A(n)
  s = n
  t = n + 1
  while s % t > 0
    s += t
    t += 1
  end
  t - n - 1
end
def A332542(n)
  (3..n).map{|i| A(i)}
end
p A332542(100) # Seiichi Manyama, Feb 19 2020

A082184 The a(n)-th triangular number is the sum of the n-th triangular number and the smallest triangular number possible.

Original entry on oeis.org

3, 6, 10, 6, 8, 28, 13, 10, 13, 18, 21, 16, 15, 26, 136, 21, 23, 40, 21, 23, 28, 38, 27, 31, 28, 28, 61, 36, 38, 496, 53, 36, 43, 36, 61, 46, 41, 44, 106, 51, 53, 91, 45, 49, 58, 78, 66, 52, 54, 53, 112, 66, 55, 58, 78, 62, 73, 98, 101, 76, 67, 106, 166, 66, 83, 142, 71

Offset: 2

Ralf Stephan, Apr 06 2003

nonn

a(n) is triangular if n+1 is triangular. Conjectures: partial maxima of sequence are at index i with value from A068195 and also a(i) - A082183(i) = 1, where i is in A068194.

Alois P. Heinz, Table of n, a(n) for n = 2..65536
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, A080824, index of second term is in A082183.

Partial maxima have index in A068194.

a:= proc(n) local h, j; h:= n*(n+1); for j from n+1 do
      if issqr(1+4*(j*(j+1)-h)) then return j fi od
    end:
seq(a(n), n=2..70);  # Alois P. Heinz, Jul 31 2019

a[n_] := Module[{h = n(n+1), j}, For[j = n+1, True, j++, If[IntegerQ[ Sqrt[1 + 4 (j(j+1) - h)]], Return[j]]]];
a /@ Range[2, 70] (* Jean-François Alcover, Jun 05 2020, after Maple *)

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(v", "); break)))

A332547 a(n) = largest odd divisor d < n of n*(n+1)/2.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 3, 5, 5, 3, 3, 7, 7, 5, 1, 9, 9, 5, 15, 11, 11, 3, 15, 13, 13, 21, 7, 15, 15, 1, 11, 17, 17, 21, 9, 19, 19, 15, 5, 21, 21, 11, 33, 23, 23, 3, 21, 35, 25, 39, 13, 27, 45, 35, 21, 29, 29, 15, 15, 31, 31, 21, 13, 55, 33, 17, 51, 35, 35

Offset: 1

N. J. A. Sloane, Feb 21 2020

nonn

Used by Michael J. Collins in the analysis of A082183 and A082184.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000217, A068194 (where a(n)=1), A082183, A082184, A332548, A332549 (where a(n)=3).

Join[{1},Table[Select[Divisors[(n(n+1))/2],OddQ[#]&&#Harvey P. Dale, Jul 16 2024 *)

A072522 Smallest number a(n)>a(n-1) such that T(a(n-1))+T(a(n))=T(m) for some m, a(1)=3; T(i) are the triangular numbers.

Original entry on oeis.org

3, 5, 6, 9, 13, 44, 58, 854, 896, 1194, 1253, 1469, 1628, 2170, 2403, 3205, 3366, 3395, 4370, 4454, 4676, 6234, 6545, 6873, 9165, 9624, 12833, 17110, 17450, 18524, 24012, 25025, 25652, 28242, 37657, 356750, 400894, 420938, 561250, 1347002

Offset: 1

Amarnath Murthy, Jul 31 2002

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..2401

Cf. A072521, A082183. Values of m are in A080824.

p=3; print1("3, "); for(n=2, 50, t=p*(p+1)/2; for(k=p, 10^9, u=t+k*(k+1)/2; v=floor(sqrt(2*u)); if(v*(v+1)/2==u, print1(k", "); p=k; break)))

Edited by Ralf Stephan, Mar 27 2003

A232178 Least k>=0 such that triangular(n) + k^2 is a square, or -1 if no such k exists.

Original entry on oeis.org

0, 0, 1, -1, -1, 1, 2, 6, 0, 2, 3, -1, -1, 3, 4, 1, 15, 4, 5, -1, -1, 5, 6, 20, 10, 6, 7, -1, -1, 7, 8, 27, 1, 8, 9, -1, -1, 9, 10, 2, 36, 10, 11, -1, -1, 11, 12, 41, 7, 0, 13, -1, -1, 13, 6, 24, 2, 14, 15, -1, -1, 15, 16, 3, 6, 8, 17, -1, -1, 17, 18, 62, 64, 18, 19

Offset: 0

Alex Ratushnyak, Nov 20 2013

sign

Triangular(n) = n*(n+1)/2.

			a(7) = 6 because the least k such that triangular(n) + k^2 is a square is k=6: 7*(7+1)/2 + 6^2 = 28+36 = 64 = 8^2.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A000217, A000290.
Cf. A082183 (least k>0 such that triangular(n) + triangular(k) is a triangular number).
Cf. A232177 (least k>0 such that triangular(n) + triangular(k) is a square).
Cf. A232176 (least k>0 such that n^2 + triangular(k) is a square).
Cf. A232179 (least k>=0 such that n^2 + triangular(k) is a triangular number).
Cf. A101157 (least k>0 such that triangular(n) + k^2 is a triangular number).

Join[{0}, Table[k = 0; While[k < n && ! IntegerQ[Sqrt[n*(n + 1)/2 + k^2]], k++]; If[k == n, k = -1]; k, {n, 100}]] (* T. D. Noe, Nov 21 2013 *)

from _future_ import division
from sympy import divisors
def A232178(n):
    if n == 0:
        return 0
    t = n*(n+1)//2
    ds = divisors(t)
    l, m = divmod(len(ds),2)
    if m:
        return 0
    for i in range(l-1,-1,-1):
        x = ds[i]
        y = t//x
        a, b = divmod(y-x,2)
        if not b:
            return a
    return -1 # Chai Wah Wu, Sep 12 2017

A232179 Least k >= 0 such that n^2 + triangular(k) is a triangular number.

Original entry on oeis.org

0, 0, 3, 1, 15, 2, 0, 3, 63, 4, 8, 5, 11, 6, 20, 3, 255, 8, 1, 9, 3, 10, 38, 11, 59, 12, 45, 13, 8, 14, 2, 15, 1023, 16, 59, 0, 24, 18, 66, 19, 51, 20, 3, 21, 44, 10, 80, 23, 251, 24, 42, 25, 68, 26, 4, 27, 39, 28, 101, 29, 10, 30, 108, 8, 4095, 32, 5, 33, 128

Offset: 0

Alex Ratushnyak, Nov 20 2013

nonn

Triangular(k) = k*(k+1)/2.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A000217, A000290, A001109.
Cf. A082183 (least k>0 such that triangular(n) + triangular(k) is a triangular number).
Cf. A232177 (least k>0 such that triangular(n) + triangular(k) is a square).
Cf. A232176 (least k>0 such that n^2 + triangular(k) is a square).
Cf. A101157 (least k>0 such that triangular(n) + k^2 is a triangular number).
Cf. A232178 (least k>=0 such that triangular(n) + k^2 is a square).

TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; Table[k = 0; While[! TriangularQ[n^2 + k*(k + 1)/2], k++]; k, {n, 0, 68}] (* T. D. Noe, Nov 21 2013 *)

a(n) = {my(k = 0); while (! ispolygonal(n^2 + k*(k+1)/2, 3), k++); k;} \\ Michel Marcus, Sep 15 2017

from _future_ import division
from sympy import divisors
def A232179(n):
    if n == 0:
        return 0
    t = 2*n**2
    ds = divisors(t)
    for i in range(len(ds)//2-1,-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

a(A001109(n)) = 0.

A332543 a(n) = n + A332542(n) + 1.

Original entry on oeis.org

6, 12, 20, 10, 14, 56, 24, 15, 22, 33, 39, 26, 21, 48, 272, 34, 38, 76, 28, 33, 46, 69, 40, 50, 39, 36, 116, 58, 62, 992, 96, 51, 70, 45, 111, 74, 57, 65, 205, 82, 86, 172, 55, 69, 94, 141, 112, 70, 75, 68, 212, 106, 66, 77, 133, 87, 118, 177, 183, 122

Offset: 3

Scott R. Shannon, Feb 18 2020

nonn

Note that A332542 is well-defined for all n. - N. J. A. Sloane, Feb 20 2020

			See A332542.

Seiichi Manyama, Table of n, a(n) for n = 3..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, April 2020

Cf. A332542, A332544, A082183.

Maple
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See A332542.
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cp's OEIS Frontend

A332552 a(n) = A082184(n) - A082183(n).

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A332553 a(n) = n + A082183(n) - A082184(n).

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A055527 Shortest other leg of a Pythagorean triangle with n as length of a leg.

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A332542 a(n) is the smallest k such that n+(n+1)+(n+2)+...+(n+k) is divisible by n+k+1.

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A082184 The a(n)-th triangular number is the sum of the n-th triangular number and the smallest triangular number possible.

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A332547 a(n) = largest odd divisor d < n of n*(n+1)/2.

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A072522 Smallest number a(n)>a(n-1) such that T(a(n-1))+T(a(n))=T(m) for some m, a(1)=3; T(i) are the triangular numbers.

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A232178 Least k>=0 such that triangular(n) + k^2 is a square, or -1 if no such k exists.

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A232179 Least k >= 0 such that n^2 + triangular(k) is a triangular number.

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