A225502 -id:A225502 - cp's OEIS frontend

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

Alex Ratushnyak, May 09 2013

nonn

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

			See A225502.

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

Cf. A000217, A112456, A225502.

#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;
}

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

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 *)

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

a(171) from Giovanni Resta, Jun 19 2013

A225787 Least prime number p such that p*triangular(n) is a triangular number, or 0 if no such p exists.

Original entry on oeis.org

2, 3, 2, 11, 19, 3, 5, 0, 71, 23, 109, 131, 17, 181, 2, 239, 271, 307, 0, 379, 3, 13, 127, 61, 67, 163, 701, 47, 811, 97, 37, 991, 0, 31, 0, 79, 83, 0, 41, 7, 0, 191, 37, 0, 5, 83, 541, 251, 2351, 613, 71, 0, 0, 2861, 743, 3079, 3191, 367, 0, 3539, 229, 0, 977, 0, 4159

Offset: 0

Alex Ratushnyak, May 16 2013

nonn

			Least prime p such that triangular(3)*p is a triangular number is p=11, so a(3) = 11.

Cf. A000217, A068084, A225502.

a(n) <= n^2 + n + 2. For n>0, a(n) <= n^n + n + 1.

A227054 a(n) = least triangular number t > 0 such that n*t is a triangular number, or 0 if no such t exists.

Original entry on oeis.org

1, 3, 1, 0, 3, 1, 3, 15, 0, 1, 6, 3, 6, 15, 1, 0, 78, 21, 10, 6, 1, 3, 45, 190, 0, 3, 55, 1, 15, 10, 15, 28203, 45, 105, 3, 1, 465, 120, 55, 3, 21, 15, 21, 3570, 1, 6, 210, 861, 0, 6, 3, 15, 105, 21945, 1, 21, 3, 66, 26565, 91, 276, 378, 6, 0, 1596, 1, 300

Offset: 1

Alex Ratushnyak, Jun 29 2013

nonn

a(n) = 1 if and only if n is a triangular number.
Indices of conjectured 0's: 4, 9, 16, 25, 49, 64, 81, 121, 144, 169, 225, 256, 289, 361, 400, 441, 529, 576, 625, 729, ... These are squares of 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27.
a(n) = 0 if n = p^(2*j) where p is a prime and j > 0. - Jon E. Schoenfield, Sep 17 2023

			a(614) = 13964154294535688630985 = A000217(167117648945) because 614 * a(614) = A000217(4141012131555), and none of the smaller triangular numbers t satisfies 614*t = A000217(m) for some m.

Cf. A166478 (indices of t in A000217), A225502, A225503.

a(1)-a(25) and a(49)-a(67) from Jon E. Schoenfield, Sep 17 2023

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

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A225787 Least prime number p such that p*triangular(n) is a triangular number, or 0 if no such p exists.

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A227054 a(n) = least triangular number t > 0 such that n*t is a triangular number, or 0 if no such t exists.

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