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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. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

LINKS

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

EXAMPLE

See A225502.

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 *)

PROG

(C)

#include <stdio.h>

#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;

}

(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

CROSSREFS

Cf. A000217, A112456, A225502.

Sequence in context: A097917 A116570 A335567 * A302350 A046879 A248267

Adjacent sequences:  A225500 A225501 A225502 * A225504 A225505 A225506

KEYWORD

nonn

AUTHOR

Alex Ratushnyak, May 09 2013

EXTENSIONS

a(171) from Giovanni Resta, Jun 19 2013

STATUS

approved

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Last modified September 20 13:50 EDT 2021. Contains 347586 sequences. (Running on oeis4.)