A125603 -id:A125603 - cp's OEIS frontend

A129445 Numbers k > 0 such that k^2 is a centered triangular number.

1, 2, 8, 19, 79, 188, 782, 1861, 7741, 18422, 76628, 182359, 758539, 1805168, 7508762, 17869321, 74329081, 176888042, 735782048, 1751011099, 7283491399, 17333222948, 72099131942, 171581218381, 713707828021, 1698478960862, 7064979148268, 16813208390239

Offset: 1

Alexander Adamchuk, Apr 15 2007, Apr 26 2007

Corresponding numbers n such that centered triangular number A005448(n) is a perfect square are listed in A129444(n).
Consider Diophantine equation 3*x*(x-1) + 2 - 2*y^2 = 0. Sequence gives solutions for y. - Zak Seidov, Jun 11 2013
Positive values of x (or y) satisfying x^2 - 10xy + y^2 + 15 = 0. - Colin Barker, Feb 09 2014
Nonnegative values of x of solutions (x, y) to the Diophantine equation 8*x^2 - 3*y^2 = 5. - Jon E. Schoenfield, Feb 02 2021

Alexander Adamchuk, Table of n, a(n) for n = 1..100
Tom Beldon and Tony Gardiner, Triangular Numbers and Perfect Squares, The Mathematical Gazette, Vol. 86, No. 507, (2002), pp. 423-431. - _Ant King_, Dec 07 2010
Index entries for linear recurrences with constant coefficients, signature (0, 10, 0, -1).

Cf. A005448, A129444.
Prime terms are listed in A129446.
Cf. A125602 (prime CTN), A184481 (semiprime CTN), A125603.
Cf. A000290, A249483.

Do[f = 3n(n-1)/2 + 1; If[IntegerQ[Sqrt[f]], Print[Sqrt[f]]], {n, 150000}]
LinearRecurrence[{0, 10, 0, -1}, {1, 2, 8, 19}, 30] (* T. D. Noe, Jun 13 2013 *)

a(n) = sqrt(3*A129444(n)*(A129444(n) - 1)/2 + 1).
G.f.: x*(1-x)*(1+3*x+x^2)/(1-10*x^2+x^4). - Colin Barker, Apr 11 2012
a(n) = 10*a(n-2) - a(n-4), a(1..4) =  1, 2, 8, 19. - Zak Seidov, Jun 11 2013

More terms from Alexander Adamchuk, Apr 26 2007

A125602 Centered triangular numbers that are prime.

Original entry on oeis.org

19, 31, 109, 199, 409, 571, 631, 829, 1489, 1999, 2341, 2971, 3529, 4621, 4789, 7039, 7669, 8779, 9721, 10459, 10711, 13681, 14851, 16069, 16381, 17659, 20011, 20359, 23251, 25939, 27541, 29191, 29611, 31321, 34429, 36739, 40099, 40591, 42589

Offset: 1

Zak Seidov, Nov 27 2006

Prime terms in A005448, or primes of the form 3n(n-1)/2 + 1.

Primes that are the sum of 3 consecutive triangular numbers. - Vicente Izquierdo Gomez, Nov 03 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A005448, A125603.

[a: n in [0..200] | IsPrime(a) where a is  (3*n^2 - 3*n + 2) div 2]; // Vincenzo Librandi, Mar 22 2013

select(isprime, [seq(3*n*(n-1)/2+1, n=1..1000)]); # Robert Israel, Nov 03 2015

lst={};Do[If[PrimeQ[p=3n(n-1)/2+1], (*Print[p];*)AppendTo[lst, p]], {n, 10^3}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
Select[Total/@Partition[Accumulate[Range[200]],3,1],PrimeQ] (* Harvey P. Dale, Dec 29 2020 *)

is(n)=n%6==1 && ispolygonal((n-1)/3,3) && isprime(n) \\ Charles R Greathouse IV, Nov 03 2015

cp's OEIS Frontend

A129445 Numbers k > 0 such that k^2 is a centered triangular number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions