A217000 Triangular numbers of the form 2p-1 where p is prime.

3, 21, 45, 105, 253, 325, 465, 561, 861, 1081, 1225, 1485, 1653, 1953, 3741, 4005, 4753, 6441, 7021, 7381, 8001, 9045, 10153, 13041, 15753, 19701, 20301, 21945, 23005, 23653, 24753, 25425, 28441, 32385, 35245, 37401, 38781, 41041, 43365, 45753, 46665, 48205

Offset: 1

Author

César Eliud Lozada, Sep 22 2012

Keywords

• nonn

Comments

Indexes n in A000217(n): A217001.

The only triangular odd number with the form 2p+1 and p prime is 15=2*7+1?

The only triangular even numbers with the form 2p and p prime are {6,10}?

From Daniel Starodubtsev, Mar 13 2020: (Start)

Proof that 15 is the only triangular number of the form 2p + 1 where p is prime: we can express T(n)=n*(n+1)/2 and p=(T(n)-1)/2=(n*(n+1)/2-1)/2=(n+2)*(n-1)/4, which can be prime only if n+2=4 or n-1=4, from which we get the only possible value n=5 (T(n)=15).

It can also be easily seen that {6,10} are the only possible values of T(n) such that T(n)/2 is prime. (End)

Examples

			For A000217 = {0, 1, 3, 6, 10, 15, 21, 28,...}, A000217(6) = 21 = 2*(11)-1. As 11 is prime then A000217(6) is in the sequence. A000217(5) = 15 = 2*(8)-1. As 8 is not prime then A000217(5) is not in the sequence.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A000217.

Cf. A124174 (2*tr+1 is also a triangular number), A217001.

Programs

Maple

tn := unapply(n*(n+1)/2,n):
f := unapply((t+1)/2,t):
T := []: N := []: P := []:
for k from 0 to 5000 do
  t:=tn(k):
  p := f(k):
  if p = floor(p) then
    p = floor(p):
    if isprime(p) then
      T := [op(T), t]:
      N := [op(N), k]:
      P := [op(P), p]:
    end if:
  end if:
  if nops(T) = 50 then
    break:
  end if:
end do:
T := T;

Mathematica

tri = 0; t = {}; Do[tri = tri + n; If[PrimeQ[(tri + 1)/2], AppendTo[t, tri]], {n, 500}]; t (* T. D. Noe, Sep 24 2012 *)