A012132 -id:A012132 - cp's OEIS frontend

A027861 Numbers k such that k^2 + (k+1)^2 is prime.

1, 2, 4, 5, 7, 9, 12, 14, 17, 19, 22, 24, 25, 29, 30, 32, 34, 35, 39, 42, 47, 50, 60, 65, 69, 70, 72, 79, 82, 84, 85, 87, 90, 97, 99, 100, 102, 104, 109, 110, 115, 122, 130, 135, 137, 139, 144, 149, 154, 157, 160, 162, 164, 167, 172, 174, 185, 187, 189, 195, 199, 202

Offset: 1

Patrick De Geest

k > 1 never ends in 1, 3, 6 or 8 (that is, k*(k+1) does not end in 2). - Lekraj Beedassy, Jul 09 2004

k > 1 can never be congruent to (1 or 3) mod 5, because if it were, then k^2 + (k+1)^2 would be divisible by 5. In other words, for k > 1, this sequence cannot contain any values in A047219. This means that we can immediately discard 40% of all possible k. - Dmitry Kamenetsky, Sep 02 2008

T. D. Noe and Zak Seidov, Table of n, a(n) for n = 1..10000
Patrick De Geest, World!Of Numbers

Complement of A012132.
Cf. A002731 (2k+1 values), A027862 (resulting primes), A091277 (indices of resulting primes).
Cf. A047219 (k mod 5 = 1 or 3), A001844 (centered squares), A010051.

a027861 n = a027861_list !! (n-1)
a027861_list = filter ((== 1) . a010051 . a001844) [0..]
-- Reinhard Zumkeller, Jul 13 2014

[n: n in [0..1000] |IsPrime(n^2 + (n+1)^2)]; // Vincenzo Librandi, Nov 19 2010

Select[Range[250],PrimeQ[#^2+(#+1)^2]&] (* Harvey P. Dale, Dec 31 2017 *)

is(n)=isprime(n^2 + (n+1)^2) \\ Charles R Greathouse IV, Apr 28 2015

a(n) = (A002731(n)-1)/2.
a(n) = (sqrt(2*A027862(n)-1)-1)/2. - Zak Seidov, Jul 22 2013
A010051(A001844(a(n))) = 1. - Reinhard Zumkeller, Jul 13 2014
a(n) = floor(sqrt(A027862(n)/2)). - Rémi Guillaume, Apr 02 2025

A089982 Triangular numbers that can be expressed as the sum of 2 positive triangular numbers.

Original entry on oeis.org

6, 21, 36, 55, 66, 91, 120, 136, 171, 210, 231, 276, 351, 378, 406, 496, 561, 666, 703, 741, 820, 861, 946, 990, 1035, 1081, 1176, 1225, 1326, 1378, 1431, 1485, 1540, 1596, 1653, 1711, 1770, 1891, 1953, 2016, 2080, 2211, 2278, 2346, 2556, 2701, 2775, 2850

Offset: 1

Jon Perry, Jan 13 2004

Intersection of triangular numbers with sumset of triangular numbers. Triangular number analog of what for squares is {A057100(n)^2} = {A009000(n)^2}. {A000217} INTERSECT {A000217 + A000217}. - Jonathan Vos Post, Mar 09 2007

A subsequence of A051533. - Wolfdieter Lang, Jan 11 2017

			Generally, A000217(A000217(n)) = A000217(A000217(n)-1) + A000217(n) and so is automatically included. These are 6=T(3), 21=T(6), 55=T(10), etc. Other solutions occur when a partial sum from x to y is triangular, e.g., 15 + 16 + 17 + 18 = 66 = T(11), so T(14) + T(11) = T(18). This particular example arises since 10+4k is triangular (at k=14, 10 + 4k = 66), and we therefore have a solution.
All other solutions occur when 3+2k, 6+3k, 10+4k, etc. -- in general, T(j) + j*k -- is triangular.

Michael S. Branicky, Table of n, a(n) for n = 1..11915

Cf. A000217, A012132, A051533, A057100.

trn[i_]:=Module[{trnos=Accumulate[Range[i]],t2s},t2s=Union[Total/@ Tuples[ trnos,2]];Intersection[trnos,t2s]] (* Harvey P. Dale, Nov 08 2011 *)
Select[Range[75], ! PrimeQ[#^2 + (# + 1)^2] &] /. Integer_ -> (Integer^2 + Integer)/2 (* Arkadiusz Wesolowski, Dec 03 2015 *)

t(i) = i*(i+1)/2;
{ v=vector(100,i,t(i)); y=vector(100); c=0; for (i=1,30, for (j=i,30, x=t(i)+t(j); f=0; for (k=1,100,if (x==v[k],f=1;break)); if (f==1,y[c++ ]=x))); select(x->(x>0), vecsort(y,,8)) } \\ slightly edited by Michel Marcus, Apr 15 2021

lista(nn) = {for (n=1, nn, my(t = n*(n+1)/2); for (k=1, n-1, if (ispolygonal(t - k*(k+1)/2, 3), print1(t, ", "); break;)););} \\ Michel Marcus, Apr 15 2021

from itertools import count, takewhile
def aupto(lim):
    t = list(takewhile(lambda x: x<=lim, (i*(i+1)//2 for i in count(1))))
    s = set(a+b for i, a in enumerate(t) for b in t[i:])
    return sorted(s & set(t))
print(aupto(3000)) # Michael S. Branicky, Jun 21 2021

Triangular number m is in this sequence iff A000161(4*m+1)>1 or, alternatively, A083025(4*m+1)>1. - Max Alekseyev, Oct 24 2008

a(n) = A000217(A012132(n)). - Ivan N. Ianakiev, Jan 17 2013

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net) and David Wasserman, Sep 23 2005

A166080 Nonprimes of the form (k^2+1)/2.

Original entry on oeis.org

1, 25, 85, 145, 221, 265, 365, 481, 545, 685, 841, 925, 1105, 1405, 1513, 1625, 1985, 2245, 2665, 2813, 2965, 3281, 3445, 3785, 3961, 4141, 4325, 4705, 4901, 5305, 5513, 5725, 5941, 6161, 6385, 6613, 6845, 7081, 7565, 7813, 8065, 8321, 8845, 9113, 9385

Offset: 1

Juri-Stepan Gerasimov, Oct 06 2009

Or, 1 together with composite numbers of the form i^2+(i+1)^2. See A012132. - N. J. A. Sloane, Feb 29 2020

A166080 U A027862 = A001844.

			a(1)=(1^2+1)/2=1; a(2)=(7^2+1)/2=25.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001844, A027862, A012132, A141468.

Select[(Range[0,150]^2+1)/2,IntegerQ[#]&&!PrimeQ[#]&] (* Harvey P. Dale, Aug 09 2025 *)

a(n) = 2n^2 + O(n^2/log n). - Charles R Greathouse IV, Mar 21 2014

Replaced 6261 by 6161 - R. J. Mathar, Oct 07 2009

A309332 Number of ways the n-th triangular number T(n) = A000217(n) can be written as the sum of two positive triangular numbers.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 2, 0, 3, 0, 0, 1, 1, 3, 0, 0, 1, 0, 1, 0, 0, 3, 1, 1, 0, 1, 3, 0, 1, 1, 1, 2, 0, 1, 2, 0, 1, 1, 2, 1, 1, 1, 1, 2, 1, 0, 3, 1, 1, 1, 0, 3, 1, 1, 0, 0, 2, 0, 1, 1, 1, 1, 1, 5, 0, 1, 1, 0, 1, 0, 0, 3, 0, 3, 1, 0, 3, 1, 3, 1, 3, 3, 0, 1, 0, 0, 3, 0, 2, 0, 1

Offset: 1

Alois P. Heinz, Aug 01 2019

nonn

The order doesn't matter. 21 = 6+15 = 15+6 are not counted as distinct solutions. - N. J. A. Sloane, Feb 22 2020

			a(3) = 1: 2*3/2 + 2*3/2 = 3*4/2.
a(21) = 2: 6*7/2 + 20*21/2 = 12*13/2 + 17*18/2 = 21*22/2.
a(23) = 3: 9*10/2 + 21*22/2 = 11*12/2 + 20*21/2 = 14*15/2 + 18*19/2 = 23*24/2.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000217, A001652, A012132, A027861, A046080 (the same for squares), A053141, A062301 (the same for primes), A108769, A309507.

a:= proc(n) local h, j, r, w; h, r:= n*(n+1), 0;
      for j from n-1 by -1 do w:= j*(j+1);
        if 2*w

a[n_] := Module[{h = n(n+1), j, r = 0, w}, For[j = n-1, True, j--, w = j(j+1); If[2w < h, Break[]]; If[ IntegerQ[Sqrt[4(h-w)+1]], r++]]; r];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Nov 16 2022, after Alois P. Heinz *)

a(n) > 0 <=> n in { A012132 }.
a(n) = 0 <=> n in { A027861 }.
a(n) = 1 <=> n in { A108769 }.

A308395 Numbers y such that x(x+1) + y(y+1) = z*(z+1) is solvable in positive integers x, z with x <= y.

Original entry on oeis.org

2, 5, 6, 9, 10, 13, 14, 17, 18, 20, 21, 22, 24, 25, 26, 27, 29, 30, 33, 34, 35, 37, 38, 39, 41, 42, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 61, 62, 65, 66, 68, 69, 70, 73, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 89, 90, 91, 92, 93, 94, 95, 97

Offset: 1

Ralf Steiner, Jul 31 2019

nonn

			14 is a term because 14*15 + 14*15 = 20*21.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A012132.

max = 220; lst = {}; For[x = 1, x < max, x++,
For[y = x, y < max, y++,
  For[z = y, z < max, z++,
   If[x (x + 1) + y (y + 1) == z (z + 1),
    lst = AppendTo[lst, y]]]]]; Select[Union[lst], # < max/2 &]

from sympy import integer_nthroot
A308395_list, y, w = [], 1, 0
while len(A308395_list) < 10000:
    w += y
    z = 0
    for x in range(1,y+1):
        z += x
        if integer_nthroot(8*(w+z)+1,2)[1]:
            A308395_list.append(y)
            break
    y += 1 # Chai Wah Wu, Aug 02 2019

A134002 Positive integers n such that n(n+5)=a(a+5)+b(b+5) is solvable in positive integers.

Original entry on oeis.org

5, 10, 11, 13, 15, 16, 20, 23, 24, 25, 30, 31, 33, 35, 36, 37, 38, 40, 42, 45, 46, 47, 49, 50, 55, 57, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 75, 76, 80, 81, 84, 85, 86, 88, 89, 90, 92, 95, 98, 99, 100, 101, 102, 105, 108, 110, 111, 112, 114, 115, 118, 120, 124, 125

Offset: 1

John W. Layman, Oct 01 2007

nonn

Conjecture. If n a positive integer not a term of this sequence, then n^2+(n+5)^2 is prime. (This has been verified up to n=500.) Examples. For n=1,2,3,4,6,7, n^2+(n+5)^2 is 37, 53,73, 97, 157 and 193, each of which is prime. See A134003 for the complement of this sequence.

			5(5+5)=50=14+36=2(2+5)+4(4+5), so 5 is a term of the sequence.

Cf. A012132, A027861, A134003.

A134003 Positive integers n for which n^2+(n+5)^2 is prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 17, 18, 19, 21, 22, 26, 27, 28, 29, 32, 34, 39, 41, 43, 44, 48, 51, 52, 53, 54, 56, 58, 59, 66, 72, 74, 77, 78, 79, 82, 83, 87, 91, 93, 94, 96, 97, 103, 104, 106, 107, 109, 113, 116, 117, 119, 121, 122, 123, 126, 134, 136, 137, 144, 151, 157

Offset: 1

John W. Layman, Oct 01 2007

nonn

Conjecture. If n is in this sequence then n(n+5)=a(a+5)+b(b+5) is not solvable in integers. (This has been verified up to n=500.) See A134002 for the complement of this sequence (in the positive integers).

Cf. A012132, A027861, A134002.

Select[Range[1,200],PrimeQ[#^2+(#+5)^2]&] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

A309388 Numbers y such that x(x+1) + y(y+1) = z*(z+1) does not have a solution in positive integers x, z with x <= y.

Original entry on oeis.org

1, 3, 4, 7, 8, 11, 12, 15, 16, 19, 23, 28, 31, 32, 36, 40, 43, 47, 52, 59, 60, 63, 64, 67, 71, 72, 79, 83, 87, 88, 96, 100, 103, 107, 108, 112, 127, 128, 131, 136, 139, 148, 151, 156, 163, 167, 172, 176, 179, 180, 183, 187, 191, 192, 196, 199, 211, 223, 227

Offset: 1

Ralf Steiner, Aug 02 2019

nonn

The similar sequence A027861 (complement of A012132) is related to primes.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from Robert Israel)

Complement of A308395.

Cf. A012132, A027861.

filter:= proc(y) local S;
  S:= map(t -> subs(t, x), [isolve(x*(x+1)+y*(y+1)=z*(z+1))]);
  select(t -> t>0 and t<=y, S) = []
end proc:
select(filter, [$1..300]); # Robert Israel, Aug 06 2019

max = 500; lst = {}; For[x = 1, x < max, x++,
For[y = x, y < max, y++,
  For[z = y, z < max, z++,
   If[x (x + 1) + y (y + 1) == z (z + 1),
    lst = AppendTo[lst, y]]]]]; lst =
Select[Union[lst], # < max/2 &]; Complement[Range[Length[lst]], lst]

from sympy import integer_nthroot
A309388_list, y, w = [], 1, 0
while len(A309388_list) < 10000:
    w += y
    z = 0
    for x in range(1,y+1):
        z += x
        if integer_nthroot(8*(w+z)+1,2)[1]:
            break
    else:
        A309388_list.append(y)
    y += 1 # Chai Wah Wu, Aug 07 2019

A108760 Irregular array: n-th row consists of nonnegative integers i less than n such that n divides i(i+1).

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 0, 4, 0, 2, 3, 5, 0, 6, 0, 7, 0, 8, 0, 4, 5, 9, 0, 10, 0, 3, 8, 11, 0, 12, 0, 6, 7, 13, 0, 5, 9, 14, 0, 15, 0, 16, 0, 8, 9, 17, 0, 18, 0, 4, 15, 19, 0, 6, 14, 20, 0, 10, 11, 21, 0, 22, 0, 8, 15, 23, 0, 24, 0, 12, 13, 25, 0, 26, 0, 7, 20, 27, 0, 28, 0, 5, 9, 14, 15, 20, 24, 29

Offset: 2

Robert Phillips (bobp(AT)usca.edu), Jun 24 2005

Row n starts with 0 and ends with n-1.
Row n of this irregular array can be viewed as the first row of an infinite matrix with elements a_{j,i} = T(n,i)+n*j. That matrix consists of all nonnegative integers i such that n divides i(i+1).
I use these matrices to generate subsequences of A012132, as you may see on page 9 of my referenced work.

			Row 12 is 0,3,8,11 which is the first row of the matrix:
   0  3  8 11
  12 15 20 23
  24 27 32 35
  ...
giving all nonnegative integers i such that 12 divides i(i+1) (cf. A108752).
Array begins:
  0, 1;
  0, 2;
  0, 3;
  0, 4;
  0, 2, 3, 5;
  0, 6;
  0, 7;
  0, 8;
  0, 4, 5, 9;
  ...

Robert Phillips, Triangular Numbers which are sums of two Triangular Numbers

Cf. A012132, A108752.

[i for n in range(2, 30) for i in range(0, n) if i*(i+1)%n==0] # Andrey Zabolotskiy, Mar 19 2022

Edited by Andrey Zabolotskiy, Mar 19 2022

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A027861 Numbers k such that k^2 + (k+1)^2 is prime.

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A089982 Triangular numbers that can be expressed as the sum of 2 positive triangular numbers.

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A166080 Nonprimes of the form (k^2+1)/2.

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A309332 Number of ways the n-th triangular number T(n) = A000217(n) can be written as the sum of two positive triangular numbers.

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A308395 Numbers y such that x*(x+1) + y*(y+1) = z*(z+1) is solvable in positive integers x, z with x <= y.

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A134002 Positive integers n such that n(n+5)=a(a+5)+b(b+5) is solvable in positive integers.

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A134003 Positive integers n for which n^2+(n+5)^2 is prime.

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A309388 Numbers y such that x*(x+1) + y*(y+1) = z*(z+1) does not have a solution in positive integers x, z with x <= y.

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A308395 Numbers y such that x(x+1) + y(y+1) = z*(z+1) is solvable in positive integers x, z with x <= y.

A309388 Numbers y such that x(x+1) + y(y+1) = z*(z+1) does not have a solution in positive integers x, z with x <= y.