A125602 -id:A125602 - 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

A276261 Centered 21-gonal primes.

Original entry on oeis.org

127, 211, 757, 2521, 2857, 6301, 8527, 16381, 19867, 23689, 24697, 27847, 32341, 37171, 38431, 42337, 66361, 68041, 82237, 89839, 97777, 103951, 114661, 140071, 152461, 162751, 170689, 192781, 204331, 216217, 231547, 240997, 284131, 308827, 353557, 357421, 385057, 389089

Offset: 1

Ilya Gutkovskiy, Aug 26 2016

nonn

Primes of the form (21*k^2 + 21*k + 2)/2.

Numbers k such that (21*k^2 + 21*k + 2)/2 is prime: 3, 4, 8, 15, 16, 24, 28, 39, 43, 47, 48, 51, 55, 059, 60, 63, 79, 80, 88, 92, 96, 99, ...

OEIS Wiki, Centered polygonal numbers
Eric Weisstein's World of Mathematics, Centered Polygonal Number
Index entries for sequences related to centered polygonal numbers

Cf. A000040, A069178.

Cf. similar sequences of the centered k-gonal primes: A125602 (k = 3), A027862 (k = 4), A145838 (k = 5), A002407 (k = 6), A144974 (k = 7), A090562 (k = 10), A262344 (k = 11), A262493 (k = 13), A264821 (k = 14), A264822 (k = 15), A264823 (k = 16), A264824 (k = 17), A264825 (k = 18), A264844 (k = 19), A264845 (k = 20), A201715 (k = 24).

Intersection[Table[(21 k^2 + 21 k + 2)/2, {k, 0, 1000}], Prime[Range[33000]]]

lista(nn) = for(n=1, nn, if(isprime(p=(21*n^2 + 21*n + 2)/2), print1(p, ", "))); \\ Altug Alkan, Aug 26 2016

A269414 Prime numbers that are the sum of one or more consecutive triangular numbers.

Original entry on oeis.org

3, 19, 31, 83, 109, 199, 251, 409, 571, 631, 683, 829, 1091, 1489, 1999, 2341, 2531, 2971, 3529, 4621, 4789, 5051, 7039, 7211, 7669, 8779, 9721, 10459, 10711, 11171, 13681, 14851, 15131, 16069, 16381, 16883, 17659, 18731, 20011, 20359, 21683, 23251, 24851

Offset: 1

Vaclav Kotesovec, Feb 25 2016

nonn

Vaclav Kotesovec and Chai Wah Wu, Table of n, a(n) for n = 1..5000 (terms 1..100 from Vaclav Kotesovec)

Cf. A257974, A000217, A125602, A034706.

A125603 Numbers n such that 3n(n-1)/2 + 1 is prime.

Original entry on oeis.org

4, 5, 9, 12, 17, 20, 21, 24, 32, 37, 40, 45, 49, 56, 57, 69, 72, 77, 81, 84, 85, 96, 100, 104, 105, 109, 116, 117, 125, 132, 136, 140, 141, 145, 152, 157, 164, 165, 169, 172, 181, 185, 189, 192, 196, 204, 205, 216, 217, 220, 221, 224, 245, 257, 264, 269, 272, 277

Offset: 1

Zak Seidov, Nov 27 2006

nonn

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

Cf. A005448, A125602.

Select[Range[300],PrimeQ[3# (#-1)/2+1]&] (* Harvey P. Dale, Mar 09 2017 *)

isok(n) = isprime(3*n*(n-1)/2 + 1); \\ Michel Marcus, Oct 11 2013

A184481 Semiprime centered triangular numbers.

Original entry on oeis.org

4, 10, 46, 85, 166, 235, 274, 361, 514, 694, 901, 1135, 1219, 1306, 1585, 1891, 2461, 2839, 3106, 3385, 3826, 3979, 4135, 5311, 5674, 6049, 6241, 6835, 7246, 8551, 9481, 10966, 11485, 11749, 12286, 12559, 13969, 15151, 15454, 17335, 18649, 18985, 19666, 21421, 21781, 22879, 23626, 24385, 26734, 27949, 28774, 30034, 32194, 33079, 33526

Offset: 1

Jonathan Vos Post, Feb 12 2011

Numbers of the form 3*n*(n-1)/2 + 1 = p*q where p and q are primes, not necessarily distinct. This is to semiprimes A001358 as A125602 is to primes A000040.

			a(3) = 3*6(6-1)/2 + 1 = 10 = 2 * 5.

Cf. A001358, A005448, A125602.

SemiprimeQ[n_] := Total[FactorInteger[n]][[2]] == 2; Select[Table[3*n (n - 1)/2 + 1, {n, 150}], SemiprimeQ]
Select[Table[(3n(n-1))/2+1,{n,200}],PrimeOmega[#]==2&] (* Harvey P. Dale, May 13 2012 *)

A001358 INTERSECTION A005448.

A257974 Prime numbers that are not the sum of one or more consecutive triangular numbers.

Original entry on oeis.org

2, 5, 7, 11, 13, 17, 23, 29, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 89, 97, 101, 103, 107, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 211, 223, 227, 229, 233, 239, 241, 257, 263, 269, 271, 277, 281, 283

Offset: 1

Vicente Izquierdo Gomez, Nov 05 2015

nonn

Subsequence of primes of A050941. - Michel Marcus, Dec 14 2015

Prime numbers that are not the difference of two tetrahedral numbers (A000292). - Franklin T. Adams-Watters, Dec 16 2015

			From _Michael De Vlieger_, Nov 06 2015: (Start)
3 is a triangular number thus is not a term.
The triangular numbers <= 7 are {1, 3, 6}. None of these are 7. 7 is not found among the sums of adjacent pairs of terms, i.e., {{1, 3}, {3, 6}} = {4, 9}. The sum of all numbers {1, 3, 6} = 10. Thus 7 is a term.
The triangular numbers <= 19 are {1, 3, 6, 10, 15}. 19 is not a triangular number. 19 is not found among sums of pairs of adjacent terms {4, 9, 16, 25} nor among those of quartets of adjacent terms {20, 34}, but is found among sums of triples of adjacent terms {10, 19, 31}. Thus 19 is not a term. (End)

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

Cf. A050941, A000217, A000292, A125602, A269414.

isA257974 := proc(n)
    if isprime(n) then
        return not isA034706(n) ;
    else
        false ;
    end if;
end proc:
for n from 0 to 400 do
    if isA257974(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 14 2015

t = Array[Binomial[# + 1, 2] &, {10^4}]; fQ[n_] := Block[{s}, s = TakeWhile[t, # <= n &]; AnyTrue[Flatten[Total /@ Partition[s, #, 1] & /@ Range[Length@ s - 1]], # == n &]]; Select[Prime@ Range@ 120, ! fQ@ # &] (* Michael De Vlieger, Nov 06 2015, Version 10 *)

More terms from Michael De Vlieger, Nov 06 2015

A307493 Primes that are both centered triangular and centered square.

Original entry on oeis.org

16381, 23199907725541, 873105326726527441, 169377932722437899461, 532026300937919058017204151243671297356368598920355705257429996547710782877327451810988538831181

Offset: 1

Ilya Gutkovskiy, Apr 10 2019

nonn

Primes that are the sum of three consecutive triangular numbers and the sum of two consecutive squares.

The next term is too large to include.

Eric Weisstein's World of Mathematics, Centered Triangular Number
Eric Weisstein's World of Mathematics, Centered Square Number

Intersection of A027862 and A125602.

Cf. A000040, A001110, A001844, A005448, A131750, A163251, A269414.

Select[LinearRecurrence[{195, -195, 1}, {1, 85, 16381}, 43], PrimeQ[#] &]

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

A276261 Centered 21-gonal primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A269414 Prime numbers that are the sum of one or more consecutive triangular numbers.

Views

Author

Keywords

Links

Crossrefs

A125603 Numbers n such that 3n(n-1)/2 + 1 is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A184481 Semiprime centered triangular numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A257974 Prime numbers that are not the sum of one or more consecutive triangular numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A307493 Primes that are both centered triangular and centered square.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica