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A217000 Triangular numbers of the form 2p-1 where p is prime. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Indexes n in A000217(n): 2, 6, 9, 14, 22, 25, 30, 33, 41, 46, 49, 54, 57, 62, 86, 89, 97, 113, 118, 121, 126, 134, 142, 161, 177, 198, 201, 209, 214, 217, 222, 225, 238, 254, 265, 273, 278, 286, 294, 302, 305, 310, 313, 321, 329, 337, 342, 350, 366, 393,...

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)

LINKS

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

EXAMPLE

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.

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 *)

CROSSREFS

Subsequence of A000217.

Cf. A124174 (Sophie Germain triangular numbers tr: 2*tr+1 is also a triangular number).

Sequence in context: A056377 A146712 A318549 * A034186 A318211 A219069

Adjacent sequences:  A216997 A216998 A216999 * A217001 A217002 A217003

KEYWORD

nonn

AUTHOR

César Eliud Lozada, Sep 22 2012

STATUS

approved

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Last modified May 18 19:38 EDT 2021. Contains 344002 sequences. (Running on oeis4.)