A225503 Least triangular number t such that t = prime(n)*triangular(m) for some m>0, or 0 if no such t exists.

6, 3, 15, 21, 66, 78, 1326, 190, 1035, 435, 465, 17205, 861, 903, 9870, 5565, 1567335, 16836, 20100, 2556, 2628, 49770, 55278, 4005, 42195, 413595, 47895, 10100265, 5995, 1437360, 32131, 8646, 1352190, 19559385, 54397665, 1642578, 12246, 52975, 501501, 134940, 336324807802305

Offset: 1

Author

Alex Ratushnyak, May 09 2013

Keywords

• nonn

Comments

Conjecture: a(n) > 0.

a(n) = (x^2-1)/8 where x is the least odd solution > 1 of the Pell-like equation x^2 - prime(n)*y^2 = 1 - prime(n). - Robert Israel, Jan 08 2015

Examples

			See A225502.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1000 (n = 1..200 from Zak Seidov).

Crossrefs

Cf. A000217, A112456, A225502.

Programs

C

#include 
#define TOP 300
typedef unsigned long long U64;
U64 isTriangular(U64 a) {
    U64 sr = 1ULL<<32, s, b, t;
    if (a < (sr/2)*(sr+1))  sr>>=1;
    while (a < sr*(sr+1)/2)  sr>>=1;
    for (b = sr>>1; b; b>>=1) {
        s = sr+b;
        if (s&1) t = s*((s+1)/2);
        else     t = (s/2)*(s+1);
        if (t >= s && a >= t)  sr = s;
    }
    return (sr*(sr+1)/2 == a);
}
int main() {
  U64 i, j, k, m, tm, p, pp = 1, primes[TOP];
  for (primes[0]=2, i = 3; pp < TOP; i+=2) {
    for (p = 1; p < pp; ++p) if (i%primes[p]==0) break;
    if (p==pp) {
        primes[pp++] = i;
        for (j=p=primes[pp-2], m=tm=1; ; j=k, m++, tm+=m) {
           if ((k = p*tm) < j) k=0;
           if (isTriangular(k)) break;
        }
        printf("%llu, ", k);
    }
  }
  return 0;
}

Maple

F:= proc(n) local p, S,x,y, z, cands, s;
      p:= ithprime(n);
      S:= {isolve(x^2 - p*y^2 = 1-p)};
      for z from 0 do
        cands:= select(s -> (subs(s,x) > 1 and subs(s,x)::odd), simplify(eval(S,_Z1=z)));
        if cands <> {} then
           x:= min(map(subs,cands, x));
           return((x^2-1)/8)
        fi
      od;
end proc:
map(F, [$1..100]); # Robert Israel, Jan 08 2015

Mathematica

a[n_] := Module[{p, x0, sol, x, y}, p = Prime[n]; x0 = Which[n == 1, 7, n == 2, 5, True, sol = Table[Solve[x > 1 && y > 1 && x^2 - p y^2 == 1 - p, {x, y}, Integers] /. C[1] -> c, {c, 0, 1}] // Simplify; Select[x /. Flatten[sol, 1], OddQ] // Min]; (x0^2 - 1)/8];
Array[a, 171] (* Jean-François Alcover, Apr 02 2019, after Robert Israel *)

PARI

a(n) = {p = prime(n); k = 1; while (! ((t=k*(k+1)/2) && ((t % p) == 0) && ispolygonal(t/p, 3)), k++); t;} \\ Michel Marcus, Jan 08 2015

Extensions

a(171) from Giovanni Resta, Jun 19 2013