A055472 -id:A055472 - cp's OEIS frontend

A124199 Primes of the form k(k+1)/2-2 (i.e., two less than triangular numbers).

13, 19, 43, 53, 89, 103, 151, 229, 251, 349, 433, 463, 593, 701, 739, 859, 1033, 1223, 1429, 1483, 1709, 1889, 1951, 2143, 2699, 3001, 3079, 3319, 3739, 4003, 4093, 4463, 4751, 5563, 5669, 6553, 7019, 7873, 8513, 9043, 10009, 10151, 10729, 11173, 11779

Offset: 1

Peter Pein (petsie(AT)dordos.net), Dec 07 2006

Equal to primes of the form (k^2-17)/8. Also equal to primes p such that 8*p+17 is a square. - Chai Wah Wu, Jul 14 2014

			The (first five triangular numbers)-2 are: -1,1,4,8,13. So a(1)=13 is the first prime of this form.

Chai Wah Wu, Table of n, a(n) for n = 1..2000.

Cf. A055472.

Pick[ #1, PrimeQ[ #1]]&[((1/2)*#1*(#1 + 1) - 2 & ) /@ Range[180]]
Select[Accumulate[Range[250]]-2,PrimeQ] (* Harvey P. Dale, Jun 07 2020 *)

isok(p) = isprime(p) && ispolygonal(p+2, 3); \\ Michel Marcus, Sep 19 2022

import sympy
[n*(n+1)/2-2 for n in range(10**6) if isprime(n*(n+1)/2-2)] # Chai Wah Wu, Jul 14 2014

A159047 Primes which are triangular numbers plus 3.

Original entry on oeis.org

3, 13, 31, 139, 193, 409, 499, 823, 1381, 1543, 2083, 2281, 3163, 3919, 6673, 7753, 9319, 9733, 17581, 19309, 21739, 22369, 24979, 27031, 27733, 30631, 39343, 40189, 51043, 53959, 54949, 57973, 62131, 67531, 70879, 81409, 85081, 86323, 91381

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 03 2009

nonn

For n>1, a(n)== 1 (mod 6). [Proof: the triangular numbers are {0,1,3,4} (mod 6), see A104686. 3 plus triangular numbers in the same set, and only those == 1 (mod 6) can be primes.] - Zak Seidov, Oct 16 2015

			13=10+3, 31=28+3, 139=136+3, 193=190+3, 409=406+3, ...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000217, A055469, A055472.

s=0;lst={};Do[s+=n;p=s+3;If[PrimeQ[p],AppendTo[lst,p]],{n,0,7!}];lst
Select[Table[n*(n + 1)/2 + 3, {n, 0, 250}], PrimeQ] (* G. C. Greubel, Jul 13 2017 *)
Select[Accumulate[Range[0,500]]+3,PrimeQ] (* Harvey P. Dale, Jul 30 2018 *)

for(n=0, 1e3, if(isprime(k=3+n*(n+1)/2), print1(k", "))) \\ Altug Alkan, Oct 16 2015

A159048 Primes of the form m*(m+1)/2 + 4.

Original entry on oeis.org

5, 7, 19, 59, 109, 157, 257, 439, 599, 907, 1039, 1229, 1279, 1489, 1657, 3407, 3659, 4099, 5569, 6907, 7507, 7879, 8389, 9049, 10589, 11329, 11939, 14369, 16657, 17209, 17959, 18149, 18919, 19507, 20507, 22159, 26339, 30139, 31379, 34457, 36319

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 03 2009

nonn

			7=3+4, 19=15+4, 59=55+4, 109=105+4, 157=153+4, 257=253+4, ...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000217, A055469, A055472, A159047

Select[Accumulate[Range[500]]+4,PrimeQ]  (* Harvey P. Dale, Apr 23 2011 *)

for(n=1, 500, if(isprime(k=n*(n+1)/2 + 4), print1(k, ", "))) \\ G. C. Greubel, Jul 03 2017

Edited by N. J. A. Sloane, Apr 06 2009

A159049 Primes of the form (5+ a triangular number A000217).

Original entry on oeis.org

5, 11, 41, 71, 83, 281, 383, 1181, 1601, 2351, 2633, 3491, 3833, 4283, 5783, 6221, 6791, 8783, 10301, 10883, 11633, 12251, 14033, 15581, 18341, 26111, 26801, 30881, 31883, 34721, 38231, 41333, 41621, 42491, 43961, 46061, 47591, 53633, 60383

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 03 2009

			11 is in the list because it is A000217(3)+5 and a prime. 41=36+5= A000217(8)+5 is a prime. 71=66+5=A000217(11)+5 is a prime.

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

Cf. A055469, A055472, A159047, A159048

s=0;lst={};Do[s+=n;p=s+5;If[PrimeQ[p],AppendTo[lst,p]],{n,0,7!}];lst
Select[Accumulate[Range[0,500]]+5,PrimeQ] (* Harvey P. Dale, Jul 08 2017 *)

for(n=1,500, if(isprime(k=n*(n+1)/2 + 5), print1(k, ", "))) \\ G. C. Greubel, Jul 13 2017

Definition rephrased, R. J. Mathar, Apr 05 2009

A185510 Array of primes in the natural number array A000027, by antidiagonals.

Original entry on oeis.org

2, 7, 3, 11, 5, 13, 29, 17, 31, 19, 37, 23, 139, 59, 41, 67, 47, 193, 109, 71, 61, 79, 107, 409, 157, 83, 97, 43, 137, 173, 499, 257, 281, 331, 73, 53, 191, 233, 823, 439, 383, 601, 127, 113, 199, 211, 353, 1381, 599, 1181, 709, 197, 179, 829, 101, 277, 467, 1543, 907, 1601, 1087, 283, 239, 1549, 163, 89

Offset: 1

Clark Kimberling, Jan 29 2011

Start with the natural number array A000027:
...2.....4....7...11...16...22...29...
...5.....8...12...17...23...30...38...
...9....13...18...24...31...39...48...
..14...19...25...32...40...49...59...
..20...26...33...41...50...60...71...
..27...34...42...51...61...72...84...
..35...43...52...62...73...85...98...
Row n of A185510 shows the primes in row n of A000027:
...7....11...29...37....67....79...137...(A055469)
...5....17...23...47...107...173...233...(A055472)
.31...139..193..409...499...823..1381...(A159047)
.59...109..157..257...439...599...907...(A159048)
.71....83..281..383..1181..1601..2351...(A159049)
.97...331..601..709..1087..1231..2707...
.73...127..197..283..307...503...673...
Conjecture:  Every row contains infinitely many primes.
Every prime occurs exactly once; that is, every prime is uniquely expressible as (1/2)(n^2 + (2k-1)n + (k-2)(k-1)) for some positive integers n and k.  We assume as true the conjecture that each row is infinite. - Clark Kimberling, Mar 10 2020

Cf. A000027, A000040, A185511, A057060, A057061.

f[n_,k_]:=n+(k+n-2)(k+n-1)/2;
TableForm[Map[Select[#,PrimeQ]&, Table[f[n,k],{n,1,20}, {k,1,100}]]]

A221055 Primes of the form k(k+1)(k+2)/6+2 (i.e., two more than a tetrahedral number).

Original entry on oeis.org

2, 3, 37, 167, 457, 971, 2927, 6547, 12343, 16217, 26237, 105997, 121487, 246907, 273821, 400997, 562477, 657361, 708563, 939931, 1072447, 1216867, 1293701, 1456937, 2027797, 2135447, 2604127, 2997413, 4410551, 5564323, 6209897, 6435691, 7647061, 8442107

Offset: 1

Jayanta Basu, Apr 15 2013

nonn

Cf. A000040, A000292, A055472.

Select[Accumulate[Table[n*(n + 1)/2, {n, 0, 2000}]] + 2, PrimeQ]

cp's OEIS Frontend

A124199 Primes of the form k(k+1)/2-2 (i.e., two less than triangular numbers).

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A159047 Primes which are triangular numbers plus 3.

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A159048 Primes of the form m*(m+1)/2 + 4.

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A159049 Primes of the form (5+ a triangular number A000217).

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A185510 Array of primes in the natural number array A000027, by antidiagonals.

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Author

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Comments

Crossrefs

Programs

Mathematica

A221055 Primes of the form k*(k+1)*(k+2)/6+2 (i.e., two more than a tetrahedral number).

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Programs

Mathematica

A221055 Primes of the form k(k+1)(k+2)/6+2 (i.e., two more than a tetrahedral number).