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Primes p such that p^2-1 is a triangular number.
2

%I #15 Dec 15 2017 17:36:09

%S 2,11,23,373,12671,901273,19472752251611,

%T 53072032161200090602953513048447623,

%U 5027153581127740201460650182713355379768873,11604855412241025458500993236724193227031777965785837784548351709747881343573

%N Primes p such that p^2-1 is a triangular number.

%C Equivalently, primes in A006452.

%C The sequence of corresponding triangular numbers begins 3, 120, 528, 139128, 160554240, 812293020528, 379188080252621270252095320, ... [_Shreevatsa R_, Jul 12 2013]

%H Joerg Arndt, <a href="/A078699/b078699.txt">Table of n, a(n) for n = 1..14</a>

%t a[n_] := a[n]=If[n<4, {1, 2, 4, 11}[[n+1]], 6a[n-2]-a[n-4]]; Select[a/@Range[200], ProvablePrimeQ] (* First do <<NumberTheory`PrimeQ` *)

%o (PARI) default(primelimit,10^7) istri(n) = t=floor(sqrt(2*n)); if(2*n==t*(t+1),1,0) forprime(p=2,5*10^6,if(istri(p^2-1),print1(p" ")))

%o (PARI) istriang(n)=issquare(8*n+1);

%o forprime(p=2,10^10,if(istriang(p^2-1),print1(p,", ")));

%o \\ _Joerg Arndt_, Jul 15 2013

%o (PARI) /* much more efficient: */

%o N=1166; f=( 1+x-4*x^2-2*x^3 ) / ( (x^2+2*x-1)*(x^2-2*x-1) )+O(x^N);

%o for(n=0,N-1,my(c=polcoeff(f,n)); if(isprime(c), print1(c,", ")));

%o \\ _Joerg Arndt_, Jul 15 2013

%Y Cf. A000217, A006452.

%K nonn

%O 1,1

%A _Jason Earls_, Dec 18 2002

%E Edited by _Dean Hickerson_, Dec 19 2002