A129545 -id:A129545 - cp's OEIS frontend

A055469 Primes of the form k(k+1)/2+1 (i.e., central polygonal numbers, or one more than triangular numbers).

2, 7, 11, 29, 37, 67, 79, 137, 191, 211, 277, 379, 631, 821, 947, 991, 1129, 1327, 1597, 1831, 2017, 2081, 2347, 2557, 2851, 2927, 3571, 3917, 4561, 4657, 4951, 5051, 5779, 6217, 6329, 8647, 8779, 9181, 9871, 11027, 12721, 13367, 14029, 14197, 14879

Offset: 1

Henry Bottomley, Jun 27 2000

Also primes of the form (n^2 + 7)/8. - Ray Chandler, Oct 08 2005
q=2 and q=5 are the only primes values such that q+1 is a triangular number because 8q+9 is a square for 2 and 5 only. - Benoit Cloitre, Apr 05 2002
n such that A000010(n) = A000217(k). - Giovanni Teofilatto, Jan 29 2010
It is conjectured that this sequence is infinite. - Daniel Forgues, Apr 21 2015

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

Cf. A000040, A000124, A000217, A067186, A110872, A110873, A129545.

Select[Table[(n^2 + 7)/8, {n, 400}], PrimeQ] (* Ray Chandler, Oct 08 2005 *)
Select[Accumulate[Range[400]]+1,PrimeQ] (* Harvey P. Dale, May 14 2022 *)

forprime(p=2,10^5, if ( issquare(8*p-7), print1(p, ", "))) \\ Joerg Arndt, Jul 14 2012

list(lim)=my(v=List(),p); forstep(s=3,sqrtint(lim\1*8-7),2, if(isprime(p=(s^2+7)/8), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, May 05 2020

a(n) = A000124(A067186(n)) = (A110873(n) + 7)/8. - Ray Chandler, Oct 08 2005

A226072 Primes p such that p-1 is a triangular number and p-2 is a square.

Original entry on oeis.org

2, 11, 11027, 16944049179227, 94511138700672573788068264540372768937231403134027

Offset: 1

Alex Ratushnyak, May 25 2013

nonn

Intersection of A129545 and A164055.
a(6) has 106 decimal digits: 12450353555...7512227.
a(7) has 381 decimal digits: 49394105798...1431227.
Roots of corresponding squares and triangular numbers: A226074 and A226073.

Cf. A226069-A226074, A129545, A164055.

import java.io.*;
import java.math.BigInteger;
public class A226072 {
public static void main (String[] args) throws Exception {
  try {
    BufferedReader in = new BufferedReader(
      new FileReader(new File("b164055.txt")));
    String line;
    while ((line = in.readLine()) != null) {
      BigInteger b = new BigInteger(line.split(" ")[1]);
      b = b.add(BigInteger.ONE);
      if (b.isProbablePrime(80))
        System.out.printf("%s, ", b.toString());
    }
  } catch (Exception e) {  e.printStackTrace();  }
}
}

Select[Prime[Range[1500]],OddQ[Sqrt[8(#-1)+1]]&&IntegerQ[Sqrt[#-2]]&] (* The program generates the first 3 terms of the sequence. *) (* Harvey P. Dale, Jul 20 2024 *)

A171569 Triangular numbers T such that T-2 is a prime.

Original entry on oeis.org

15, 21, 45, 55, 91, 105, 153, 231, 253, 351, 435, 465, 595, 703, 741, 861, 1035, 1225, 1431, 1485, 1711, 1891, 1953, 2145, 2701, 3003, 3081, 3321, 3741, 4005, 4095, 4465, 4753, 5565, 5671, 6555, 7021, 7875, 8515, 9045, 10011, 10153, 10731, 11175, 11781

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 11 2009

nonn

			15-2 = 13, 21-2 = 19, 45-2 = 43, ...

Cf. A000217, A129545, A129752.

Select[s=0;Table[n*(n+1)/2,{n,6!}],PrimeQ[ #-2]&]
Select[Accumulate[Range[200]],PrimeQ[#-2]&] (* Harvey P. Dale, Apr 09 2018 *)

A171570 Triangular numbers T such that T+2 is a prime.

Original entry on oeis.org

1, 3, 15, 21, 45, 105, 171, 231, 351, 465, 561, 741, 861, 1275, 1431, 1485, 2211, 2415, 2775, 3081, 3321, 4005, 4371, 5151, 7875, 8385, 10731, 11175, 11781, 13041, 13695, 14535, 15051, 15225, 17205, 17391, 17955, 18915, 21321, 22155, 23871, 24531

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 11 2009

nonn

			1+2=3, 3+2=5, 15+2=17, ..

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

Cf. A000217, A129545, A129752, A171569.

Select[s=0;Table[n*(n+1)/2,{n,6!}],PrimeQ[ #+2]&]
Select[Accumulate[Range[300]],PrimeQ[#+2]&] (* Harvey P. Dale, Jun 27 2017 *)

A345350 Even triangular numbers such that the next integer is nonprime.

Original entry on oeis.org

0, 120, 300, 406, 496, 528, 666, 780, 1176, 1378, 1540, 1770, 2278, 2628, 3160, 3240, 3486, 3828, 4186, 4278, 5356, 5460, 5886, 6670, 6786, 7140, 7260, 7626, 7750, 8128, 8256, 9316, 9730, 10296, 10440, 10878, 11476, 11628, 12090, 12246, 12880, 13530, 14706, 15576

Offset: 1

Tanya Khovanova, Jun 15 2021

nonn

Subsequence of A000217 (triangular numbers) and A014494 (even triangular numbers).

			Even triangular numbers 6, 10, 28, 36, 66, and 78 are all followed by a prime number. Even triangular number 120 is followed by a composite number 121. Thus, a(1) = 120.

Cf. A000217, A014494, A129545.

Select[Table[n (n + 1)/2, {n, 0, 200}], EvenQ[#] && ! PrimeQ[# + 1] &]
Select[Accumulate[Range[0,300]],EvenQ[#]&&!PrimeQ[#+1]&] (* Harvey P. Dale, Mar 23 2025 *)

lista(nn) = for (n=1, nn, my(t=n*(n+1)/2); if (!(t%2) && !isprime(t+1), print1(t, ", "))) \\ Michel Marcus, Jun 16 2021

from sympy import isprime
def A014494(n): return (2*n+1)*(2*n+1-(-1)**n)//2
def ok(et): return not isprime(et+1)
print(list(filter(ok, (A014494(n) for n in range(90))))) # Michael S. Branicky, Jun 15 2021

cp's OEIS Frontend

A055469 Primes of the form k(k+1)/2+1 (i.e., central polygonal numbers, or one more than triangular numbers).

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A226072 Primes p such that p-1 is a triangular number and p-2 is a square.

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A171569 Triangular numbers T such that T-2 is a prime.

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A171570 Triangular numbers T such that T+2 is a prime.

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A345350 Even triangular numbers such that the next integer is nonprime.

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