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A225502 Least m > 0 such that prime(n)*triangular(m) is a triangular number, or 0 if no such m exists. 3
2, 1, 2, 2, 3, 3, 12, 4, 9, 5, 5, 30, 6, 6, 20, 14, 230, 23, 24, 8, 8, 35, 36, 9, 29, 90, 30, 434, 10, 159, 22, 11, 140, 530, 854, 147, 12, 25, 77, 39, 1938509, 13, 41, 69, 182, 70, 14, 104, 105, 60, 30, 15, 15, 47, 240, 65274, 6314, 16, 17009, 33, 50, 68, 17, 264, 371 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture: a(n) > 0.

LINKS

Table of n, a(n) for n=1..65.

EXAMPLE

n prime(n) m tri(m) prime(n)*tri(m)

1 2 2 3 6

2 3 1 1 3

3 5 2 3 15

4 7 2 3 21

5 11 3 6 66

6 13 3 6 78

7 17 12 78 1326

8 19 4 10 190

MATHEMATICA

lm[n_]:=Module[{m=1, p=Prime[n]}, While[!OddQ[Sqrt[8(p (m(m+1))/2)+1]], m++]; m]; Array[lm, 68] (* Harvey P. Dale, Mar 16 2018 *)

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) { m=0; break; }

if (isTriangular(k)) break;

}

printf("%llu, ", m);

}

}

return 0;

}

CROSSREFS

Cf. A000217, A112456, A225503.

Sequence in context: A161281 A226916 A003113 * A152227 A246583 A244788

Adjacent sequences: A225499 A225500 A225501 * A225503 A225504 A225505

KEYWORD

nonn

AUTHOR

Alex Ratushnyak, May 09 2013

STATUS

approved

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Last modified February 6 11:28 EST 2023. Contains 360104 sequences. (Running on oeis4.)