A055527 -id:A055527 - 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

A084921 a(n) = lcm(p-1, p+1) where p is the n-th prime.

Original entry on oeis.org

3, 4, 12, 24, 60, 84, 144, 180, 264, 420, 480, 684, 840, 924, 1104, 1404, 1740, 1860, 2244, 2520, 2664, 3120, 3444, 3960, 4704, 5100, 5304, 5724, 5940, 6384, 8064, 8580, 9384, 9660, 11100, 11400, 12324, 13284, 13944, 14964, 16020, 16380

Offset: 1

Reinhard Zumkeller, Jun 11 2003

nonn

This sequence consists of terms of sequences A055523 and A055527 for prime n > 2. - Toni Lassila (tlassila(AT)cc.hut.fi), Feb 02 2004

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Christian Krause, LODA program for A084921
Index entries for sequences related to lcm's

Cf. A000040, A006093, A008864, A009286, A024700, A055523, A055527, A084920, A084922.

a084921 n = lcm (p - 1) (p + 1)  where p = a000040 n
-- Reinhard Zumkeller, Jun 01 2013

[3] cat [(p^2-1)/2: p in PrimesInInterval(3,300)]; // G. C. Greubel, May 03 2024

LCM[#-1,#+1]&/@Prime[Range[50]] (* Harvey P. Dale, Oct 09 2018 *)

a(n)=if(n<2,3,(prime(n)^2-1)/2) \\ Charles R Greathouse IV, May 15 2013

[3]+[(n^2-1)/2 for n in prime_range(3,301)] # G. C. Greubel, May 03 2024

a(n) = A084920(n)/2 for n > 1.
a(n) = 3*A084922(n) for n > 2.
a(n) = A009286(A000040(n)). - Enrique Pérez Herrero, May 17 2012
a(n) ~ 0.5 n^2 log^2 n. - Charles R Greathouse IV, May 15 2013
Product_{n>=1} (1 + 1/a(n)) = 2. - Amiram Eldar, Jan 23 2021
a(n) = (A000040(n)^2 - 1) / 2 for n > 1. - Christian Krause, Mar 27 2021
a(n) = (3/2)*A024700(n-2), for n > 1. - G. C. Greubel, May 03 2024

A055523 Longest other leg of a Pythagorean triangle with n as length of a leg.

Original entry on oeis.org

4, 3, 12, 8, 24, 15, 40, 24, 60, 35, 84, 48, 112, 63, 144, 80, 180, 99, 220, 120, 264, 143, 312, 168, 364, 195, 420, 224, 480, 255, 544, 288, 612, 323, 684, 360, 760, 399, 840, 440, 924, 483, 1012, 528, 1104, 575, 1200, 624, 1300, 675, 1404, 728, 1512, 783

Offset: 3

Henry Bottomley, May 22 2000

Todd Silvestri, Table of n, a(n) for n = 3..1002
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A009112, A046079, A046080, A046081, A054435, A054436, A055522, A055524, A055525, A055526, A055527.

seq(`if`(n::even, (n/2-1)*(n/2+1), (n-1)*(n+1)/2), n=3..100); # Robert Israel, Dec 16 2014

a[n_Integer/;n>=3]:=(3 (n^2-2)+(-1)^(n+1) (n^2+2))/8 (* Todd Silvestri, Dec 16 2014 *)

Vec(x^3*(x^3-3*x-4)/((x-1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Sep 15 2014

a(n) = 2*A055522(n)/n = sqrt(A055524(n)^2-n^2).
a(2k) = (k-1)*(k+1), a(2k+1) = 2k*(k+1).
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). G.f.: x^3*(x^3-3*x-4) / ((x-1)^3*(x+1)^3). - Colin Barker, Sep 15 2014
a(n) = (3*(n^2-2)+(-1)^(n+1)*(n^2+2))/8. - Todd Silvestri, Dec 16 2014
E.g.f.: 1 + (3*x^2/8 + 3*x/8 - 3/4)*exp(x) + (-x^2/8 + x/8 - 1/4)*exp(-x). - Robert Israel, Dec 16 2014

A055524 Longest other side of a Pythagorean triangle with n as length of one of the three sides (in fact n is a leg and a(n) the hypotenuse).

Original entry on oeis.org

5, 5, 13, 10, 25, 17, 41, 26, 61, 37, 85, 50, 113, 65, 145, 82, 181, 101, 221, 122, 265, 145, 313, 170, 365, 197, 421, 226, 481, 257, 545, 290, 613, 325, 685, 362, 761, 401, 841, 442, 925, 485, 1013, 530, 1105, 577, 1201, 626, 1301, 677, 1405, 730, 1513, 785

Offset: 3

Henry Bottomley, May 22 2000

Paolo Xausa, Table of n, a(n) for n = 3..10000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A009112, A046079, A046080, A046081, A054435, A054436, A055522, A055523, A055525, A055526, A055527.

A055524[n_] := (3*n^2-(-1)^n*(n^2-2)+6)/8; Array[A055524, 100, 3] (* or *)
LinearRecurrence[{0, 3, 0, -3, 0, 1}, {5, 5, 13, 10, 25, 17}, 100] (* Paolo Xausa, Feb 29 2024 *)

Vec(-x^3*(2*x^5+x^4-5*x^3-2*x^2+5*x+5)/((x-1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Sep 15 2014

a(n) = sqrt(n^2+A055523(n)^2). a(2k) = k^2+1, a(2k+1) = k^2+(k+1)^2.
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). G.f.: -x^3*(2*x^5+x^4-5*x^3-2*x^2+5*x+5) / ((x-1)^3*(x+1)^3). - Colin Barker, Sep 15 2014
a(n) = (3*n^2+6-(n^2-2)*(-1)^n)/8. - Luce ETIENNE, Jul 11 2015

A055525 Shortest other side of a Pythagorean triangle having n as length of one of the three sides.

Original entry on oeis.org

4, 3, 3, 8, 24, 6, 12, 6, 60, 5, 5, 48, 8, 12, 8, 24, 180, 12, 20, 120, 264, 7, 7, 10, 36, 21, 20, 16, 480, 24, 44, 16, 12, 15, 12, 360, 15, 9, 9, 40, 924, 33, 24, 528, 1104, 14, 168, 14, 24, 20, 28, 72, 33, 33, 76, 40, 1740, 11, 11, 960, 16, 48, 16, 88, 2244, 32, 92, 24

Offset: 3

Henry Bottomley, May 22 2000

nonn

Robert G. Wilson v, Table of n, a(n) for n = 3..10000

Cf. A009112, A046079, A046080, A046081, A054435, A054436, A055522, A055523, A055524, A055526, A055527.

a[n_] := Block[{a, c, k = 1, n2 = n^2}, While[ If[ k > n, !IntegerQ[c = Sqrt[n2 + k^2]], !IntegerQ[c = Sqrt[n2 + k^2]] && !IntegerQ[a = Sqrt[n2 - k^2]]], k++; If[k == n, k++]]; If[ IntegerQ@ c, k, Sqrt[n2 - a^2]]]; (* Robert G. Wilson v, Feb 23 2024 *)

From Robert G. Wilson v, Feb 23 2024: (Start)
sqrt(2*(n-1)) < a(n) < n^2/2.
If n = k*m, then a(n) <= k*a(m). (End)

A055522 Largest area of a Pythagorean triangle with n as length of one of the three sides (in fact as a leg).

Original entry on oeis.org

6, 6, 30, 24, 84, 60, 180, 120, 330, 210, 546, 336, 840, 504, 1224, 720, 1710, 990, 2310, 1320, 3036, 1716, 3900, 2184, 4914, 2730, 6090, 3360, 7440, 4080, 8976, 4896, 10710, 5814, 12654, 6840, 14820, 7980, 17220, 9240, 19866, 10626, 22770, 12144, 25944

Offset: 3

Henry Bottomley, May 22 2000

Cf. A009112, A046079, A046080, A046081, A054435, A054436, A055523, A055524, A055525, A055526, A055527.

seq(piecewise(n mod 2 = 0,n*(n^2-4)/8,n*(n^2-1)/4),n=3..60); # C. Ronaldo

Table[n*(3*(n^2 - 2) - (n^2 + 2)*(-1)^n)/16, {n, 3, 50}] (* Wesley Ivan Hurt, Apr 27 2017 *)

a(n) = n*A055523(n)/2.
a(2k) = k*(k+1)*(k-1), a(2k+1) = k*(k+1)*(2k+1).
O.g.f.: 6*x^3*(x+1+x^2)/((1-x)^4*(1+x)^4). a(2k+1)=A055112(k). a(2k)=A007531(k+1). [R. J. Mathar, Aug 06 2008]
a(n) = n*(3*(n^2-2)-(n^2+2)*(-1)^n)/16. - Luce ETIENNE, Jul 17 2015

A055526 Shortest hypotenuse of a Pythagorean triangle with n as length of a leg.

Original entry on oeis.org

5, 5, 13, 10, 25, 10, 15, 26, 61, 13, 85, 50, 17, 20, 145, 30, 181, 25, 29, 122, 265, 25, 65, 170, 45, 35, 421, 34, 481, 40, 55, 290, 37, 39, 685, 362, 65, 41, 841, 58, 925, 55, 51, 530, 1105, 50, 175, 130, 85, 65, 1405, 90, 73, 65, 95, 842, 1741, 61, 1861, 962, 65

Offset: 3

Henry Bottomley, May 22 2000

nonn

Smallest k>n such that the squarefree part of k+n equals the squarefree part of k-n - Benoit Cloitre, May 26 2002

Paolo Xausa, Table of n, a(n) for n = 3..1000

Cf. A009112, A046079, A046080, A046081, A054435, A054436, A055522, A055523, A055524, A055525, A055527.

core[n_] := core[n] = Times @@ Map[#[[1]]^Mod[#[[2]], 2] &, FactorInteger[n]];
A055526[n_] := Block[{k = n}, While[core[++k+n] != core[k-n]]; k];
Array[A055526, 100, 3] (* Paolo Xausa, Feb 29 2024 *)

for(n=3,105,s=n+1; while(abs(core(s+n)-core(s-n))>0,s++); print1(s,","))

a(n) = sqrt(n^2+A055527(n)^2).

A054435 Smallest area of a Pythagorean triangle with n as length of one of the three sides.

Original entry on oeis.org

6, 6, 6, 24, 84, 24, 54, 24, 330, 30, 30, 336, 54, 96, 60, 216, 1710, 96, 210, 1320, 3036, 84, 84, 120, 486, 294, 210, 216, 7440, 384, 726, 240, 210, 270, 210, 6840, 270, 180, 180, 840, 19866, 726, 486, 12144, 25944, 336, 4116, 336, 540, 480, 630, 1944, 726

Offset: 3

Henry Bottomley, May 22 2000

nonn

Cf. A009112, A046079, A046080, A046081, A054436, A055522, A055523, A055524, A055525, A055526, A055527.

A054436 Smallest area of a Pythagorean triangle with n as length of a leg.

Original entry on oeis.org

6, 6, 30, 24, 84, 24, 54, 120, 330, 30, 546, 336, 60, 96, 1224, 216, 1710, 150, 210, 1320, 3036, 84, 750, 2184, 486, 294, 6090, 240, 7440, 384, 726, 4896, 210, 270, 12654, 6840, 1014, 180, 17220, 840, 19866, 726, 540, 12144, 25944, 336, 4116, 3000, 1734, 1014

Offset: 3

Henry Bottomley, May 22 2000

nonn

Cf. A009112, A046079, A046080, A046081, A054435, A055522, A055523, A055524, A055525, A055526, A055527.

readlib(issqr): for a from 3 to 80 do for b from 1 by 1 while not issqr(a^2+b^2) do od: printf("%d, ",a*b/2) od: # C. Ronaldo

a[n_] := For[k = 1, True, k++, If[IntegerQ[Sqrt[n^2+k^2]], Return[n k/2]]];
a /@ Range[3, 100] (* Jean-François Alcover, Feb 14 2020 *)

a(n) = n*A055527(n)/2.

A232176 Least positive k such that n^2 + triangular(k) is a square.

Original entry on oeis.org

1, 2, 6, 10, 14, 18, 7, 5, 8, 34, 6, 42, 46, 15, 54, 16, 14, 66, 70, 74, 23, 82, 9, 90, 17, 98, 102, 10, 110, 15, 25, 122, 126, 16, 39, 48, 40, 21, 150, 34, 158, 29, 54, 48, 30, 13, 182, 63, 55, 194, 56, 202, 14, 45, 214, 63, 222, 26, 41, 234, 31, 42, 39, 250, 32, 63

Offset: 0

Alex Ratushnyak, Nov 19 2013

nonn

Triangular(k) = A000217(k) = k*(k+1)/2.
a(n) <= 4*n - 2, because with k = 4*n-2: n^2 + k*(k+1)/2 = n^2 + (4*n-2)*(4*n-1)/2 = 9*n^2 - 6*n + 1 = (3*n-1)^2.
The sequence of numbers n such that a(n)=n begins: 8, 800, 7683200 ... - a subsequence of A220186.

Andrey Zabolotskiy, Table of n, a(n) for n = 0..5000

Cf. A000290, A000217, A038202, A055527, A220186, A232175.
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).
Cf. A232178 (least k>=0 such that triangular(n) + k^2 is a square).

lpk[n_]:=Module[{k=1},While[!IntegerQ[Sqrt[n^2+(k(k+1))/2]],k++];k]; Array[ lpk,70,0] (* Harvey P. Dale, May 04 2018 *)

a(n) = {k = 1; while (! issquare(n^2 + k*(k+1)/2), k++); k;} \\ Michel Marcus, Nov 20 2013

import math
for n in range(77):
  n2 = n*n
  y=1
  for k in range(1,10000001):
    sum = n2 + k*(k+1)//2
    r = int(math.sqrt(sum))
    if r*r == sum:
      print(str(k), end=',')
      y=0
      break
  if y: print('-', end=',')

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.

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A084921 a(n) = lcm(p-1, p+1) where p is the n-th prime.

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A055523 Longest other leg of a Pythagorean triangle with n as length of a leg.

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A055524 Longest other side of a Pythagorean triangle with n as length of one of the three sides (in fact n is a leg and a(n) the hypotenuse).

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A055525 Shortest other side of a Pythagorean triangle having n as length of one of the three sides.

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A055522 Largest area of a Pythagorean triangle with n as length of one of the three sides (in fact as a leg).

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A055526 Shortest hypotenuse of a Pythagorean triangle with n as length of a leg.

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A054435 Smallest area of a Pythagorean triangle with n as length of one of the three sides.

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