A217000 - OEIS

%I #37 Oct 31 2024 06:48:37

%S 3,21,45,105,253,325,465,561,861,1081,1225,1485,1653,1953,3741,4005,

%T 4753,6441,7021,7381,8001,9045,10153,13041,15753,19701,20301,21945,

%U 23005,23653,24753,25425,28441,32385,35245,37401,38781,41041,43365,45753,46665,48205

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

%C Indexes n in A000217(n): A217001.

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

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

%C From _Daniel Starodubtsev_, Mar 13 2020: (Start)

%C 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).

%C 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)

%H T. D. Noe, <a href="/A217000/b217000.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%p tn := unapply(n*(n+1)/2,n):

%p f := unapply((t+1)/2,t):

%p T := []: N := []: P := []:

%p for k from 0 to 5000 do

%p   t:=tn(k):

%p   p := f(k):

%p   if p = floor(p) then

%p     p = floor(p):

%p     if isprime(p) then

%p       T := [op(T), t]:

%p       N := [op(N), k]:

%p       P := [op(P), p]:

%p     end if:

%p   end if:

%p   if nops(T) = 50 then

%p     break:

%p   end if:

%p end do:

%p T := T;

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

%Y Subsequence of A000217.

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

%K nonn

%O 1,1

%A _César Eliud Lozada_, Sep 22 2012