A061782 -id:A061782 - cp's OEIS frontend

A267077 Least m>0 for which mn^2 + 1 is a square and mtriangular(n) + 1 is a triangular number (A000217). Or -1 if no such m exists.

1, 35, 30, 18135, 189, 27, 321300, 23760, 1188585957, 1656083, 26, 244894427400, 82093908624206325, 1858717755529547, 86478, 21491811639746039592, 26135932603458945934958445, 353382195058506640426335, 26780050, 7859354769338288038121982384, 274554988002

Offset: 0

Alex Ratushnyak, Jan 10 2016

nonn

			26*10^2+1 = 2601 is a square, and 26*10*11/2+1 = 1431 = triangular(53), and 26 is the smallest such multiplier, therefore a(10) = 26.

Don Reble, Table of n, a(n) for n = 0..300

Cf. A000217, A000290, A035096, A061782, A067872, A188621.

from math import sqrt
def A267077(n):
    if n == 0:
        return 1
    u,v,t,w = max(8,2*n),max(4,n)**2-9,4*n*(n+1),n**2
    while True:
        m,r = divmod(v,t)
        if not r and int(sqrt(m*w+1))**2 == m*w+1:
            return m
        v += u+1
        u += 2 # Chai Wah Wu, Jan 15 2016

#!/usr/bin/python3
# This sequence is easy if you use a Pell-equation solver such as labmath.py
# Solve the A267077 Pell equation:
# nx^2 - (4n+4)y^2 = 5n-4; but also y^2 == 1 mod n^2
# Let u = nx, then # u^2 - n*(4n+4)y^2 = n*(5n-4)
#   and (y > n) and (u == 0 mod n) and (y^2 == 1 mod n^2)
# (y > n makes m > 0)
# Report m = (y^2 - 1) / n^2
import labmath
print(0, 1)
print(1, 35) # When n<2, the Pell equation is elliptical.
for nn in range(2,1001):
    nsq = nn * nn
    ps = labmath.pell(nn*(4*nn+4), nn*(5*nn-4))
    uu,yy = next(ps[0])
    while (yy <= nn) or ((uu % nn) != 0) or ((yy*yy) % nsq != 1):
        uu,yy = next(ps[0])
    print(nn, (yy*yy - 1) // nsq)
# From Don Reble, Apr 15 2022, added by N. J. A. Sloane, Apr 15 2022.

a(12)-a(15) from Chai Wah Wu, Jan 16 2016

a(16) and beyond from Don Reble, Apr 15 2022

A344006 a(n) = m(m+1)/n, where A344005(n) is the smallest number m such that n divides m(m+1).

Original entry on oeis.org

2, 1, 2, 3, 4, 1, 6, 7, 8, 2, 10, 1, 12, 3, 2, 15, 16, 4, 18, 1, 2, 5, 22, 3, 24, 6, 26, 2, 28, 1, 30, 31, 4, 8, 6, 2, 36, 9, 4, 6, 40, 1, 42, 3, 2, 11, 46, 5, 48, 12, 6, 3, 52, 13, 2, 1, 6, 14, 58, 4, 60, 15, 12, 63, 10, 2, 66, 4, 8, 3, 70, 1, 72, 18, 8, 5, 6, 2, 78, 3, 80, 20, 82, 5

Offset: 1

N. J. A. Sloane, Jun 04 2021

nonn

a(n) = n-1 if n is a prime power. - Chai Wah Wu, Jun 04 2021

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Cf. A011772, A061782, A344005.

a(n) = for(m=1, oo, if((m*(m+1))%n==0, return(m*(m+1)/n))) \\ Felix Fröhlich, Jun 04 2021
(Python 3.8+)
from itertools import combinations
from math import prod
from sympy import factorint
from sympy.ntheory.modular import crt
def A344006(n):
    if n == 1:
        return 2
    plist = [p**q for p, q in factorint(n).items()]
    if len(plist) == 1:
        return n-1
    else:
        m = int(min(min(crt([m,n//m],[0,-1])[0],crt([n//m,m],[0,-1])[0]) for m in (prod(d) for l in range(1,len(plist)//2+1) for d in combinations(plist,l))))
        return m*(m+1)//n # Chai Wah Wu, Jun 04 2021

A382244 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, n*a(n) is a triangular number (A000217).

Original entry on oeis.org

0, 1, 3, 2, 7, 9, 6, 4, 15, 5, 12, 21, 10, 25, 27, 8, 31, 33, 35, 37, 39, 11, 24, 45, 22, 13, 30, 14, 42, 57, 26, 16, 63, 17, 67, 18, 56, 19, 75, 20, 52, 81, 28, 85, 87, 23, 51, 93, 95, 97, 99, 46, 40, 105, 60, 90, 36, 29, 66, 117, 54, 121, 69, 32, 127, 84, 58

Offset: 0

Rémy Sigrist, Mar 19 2025

nonn

A self-inverse permutation of the nonnegative integers.

			The initial terms are:
  n   a(n)  n*a(n)
  --  ----  -----------------
   0     0    0 = A000217(0)
   1     1    1 = A000217(1)
   2     3    6 = A000217(3)
   3     2    6 = A000217(3)
   4     7   28 = A000217(7)
   5     9   45 = A000217(9)
   6     6   36 = A000217(8)
   7     4   28 = A000217(7)
   8    15  120 = A000217(15)
   9     5   45 = A000217(9)
  10    12  120 = A000217(15)
  11    21  231 = A000217(21)
  12    10  120 = A000217(15)
  13    25  325 = A000217(25)
  14    27  378 = A000217(27)
  15     8  120 = A000217(15)

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A000217, A061782, A382244.

PARI
```
\\ See Links section.
```

a(n) >= A061782(n) for any n > 0.

A193470 Square array A(n,k) (n>=1, k>=0) read by antidiagonals: A(n,0) = 0 and A(n,k) is the least integer > A(n,k-1) that can be expressed as a triangular number divided by n.

Original entry on oeis.org

0, 0, 1, 0, 3, 3, 0, 1, 5, 6, 0, 7, 2, 14, 10, 0, 2, 9, 5, 18, 15, 0, 1, 3, 30, 7, 33, 21, 0, 3, 6, 9, 34, 12, 39, 28, 0, 15, 4, 11, 11, 69, 15, 60, 36, 0, 4, 17, 13, 13, 21, 75, 22, 68, 45, 0, 1, 5, 62, 15, 20, 24, 124, 26, 95, 55, 0, 5, 12, 17, 66, 30, 35, 38, 132, 35, 105, 66

Offset: 1

Peter Luschny, Jul 27 2011

			n\k  0   1   2    3    4     5     6     7
------------------------------------------
1 |  0   1   3    6   10    15    21    28    A000217
2 |  0   3   5   14   18    33    39    60    A074378
3 |  0   1   2    5    7    12    15    22    A001318
4 |  0   7   9   30   34    69    75   124    A154260
5 |  0   2   3    9   11    21    24    38    A057569
6 |  0   1   6   11   13    20    35    46    A154293
7 |  0   3   4   13   15    30    33    54    A057570
8 |  0  15  17   62   66   141   147   252    A157716

Cf. A061782, A011772.

A193470_rect := proc(n,k) local j,i,L; L := NULL; j := 0; while nops([L]) < k do add(i/n, i=1..j); if type(%,integer) then L := L,% fi; j := j+1 od; L end:
seq(print(A193470_rect(n, 12)),n = 1..8);

a[, 0] = 0; a[n, k_] := a[n, k] = For[j = a[n, k-1]+1, True, j++, If[Reduce[m > 0 && j == m(m+1)/(2n), m, Integers] =!= False, Return[j]]]; Table[a[n-k, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 07 2016 *)

A344949 a(n) is the smallest square s > 0 such that s*(2n+1) is a triangular number.

Original entry on oeis.org

1, 1, 9, 4, 4, 441, 25, 1, 9, 9, 1, 3218436, 49, 1089, 1656369, 16, 16, 225, 46225, 9, 81, 314721, 1, 12217323024, 25, 25, 2427192623025, 1, 2304, 199572129, 121, 400, 81225, 39727809, 4, 36, 36, 4, 736164, 94864, 592900, 4357032433168041, 169, 3025, 3600, 1

Offset: 0

Mihai Prunescu, Jun 03 2021

nonn

Proof that every odd natural number 2n+1 is a triangular number divided by a square. As the number 4n + 2 is never a square, Pell's equation x^2 - (4n+2)*y^2 = 1 has solutions in integers with y != 0 for every n. It is immediate that x has to be odd. We replace x = 2b+1 and we observe that y must be then even. We replace y = 2a and it follows that b(b+1)/2 = (2n+1)*a^2. So (2n+1) is a triangular number divided by a square. Of course, for given n, there are infinitely many such pairs (b,a).

Cf. A000217, A000290, A061782.

Table[k=1;While[!IntegerQ[Sqrt[8k^2(2n+1)+1]],k++];k^2,{n,0,22}] (* Giorgos Kalogeropoulos, Jun 03 2021 *)

a(n) = my(k=1); while (!ispolygonal(k^2*(2*n+1), 3), k++); k^2; \\ Michel Marcus, Jun 06 2021

from sympy.solvers.diophantine.diophantine import diop_DN
def A344949(n): return min(d[1]**2 for d in diop_DN(4*n+2, 1))//4 # Chai Wah Wu, Jun 21 2021

cp's OEIS Frontend

A267077 Least m>0 for which m*n^2 + 1 is a square and m*triangular(n) + 1 is a triangular number (A000217). Or -1 if no such m exists.

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A344006 a(n) = m*(m+1)/n, where A344005(n) is the smallest number m such that n divides m*(m+1).

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A382244 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, n*a(n) is a triangular number (A000217).

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A193470 Square array A(n,k) (n>=1, k>=0) read by antidiagonals: A(n,0) = 0 and A(n,k) is the least integer > A(n,k-1) that can be expressed as a triangular number divided by n.

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A344949 a(n) is the smallest square s > 0 such that s*(2n+1) is a triangular number.

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PARI

Python

A267077 Least m>0 for which mn^2 + 1 is a square and mtriangular(n) + 1 is a triangular number (A000217). Or -1 if no such m exists.

A344006 a(n) = m(m+1)/n, where A344005(n) is the smallest number m such that n divides m(m+1).