

A246042


Numbers n such that there is both a square and a triangular number strictly between n^3 and n^3+n.


1




OFFSET

1,1


COMMENTS

Is the sequence infinite?
a(9) > 10^10.  Hiroaki Yamanouchi, Sep 03 2014
a(9) > 10^12.  Jon E. Schoenfield, Dec 23 2014
Under certain assumptions, a heuristic argument can be made that the sequence is infinite. Given a value of n, let I_n be the interval consisting of the n1 integers lying strictly between n^3 and n^3+n. The density of squares in the vicinity of a large number C is 1/(2*sqrt(C)), so if we assume that we may treat the event that I_n includes a square as a random event, its probability of occurrence should be about P_S = (n1)/(2*sqrt(n^3)), which approaches 1/(2*sqrt(n)) for large n. Similarly, the density of triangular numbers in the vicinity of a large number C is 1/sqrt(2*C), so if we assume that we may treat the event that I_n includes a triangular number as a random event, its probability of occurrence should be about P_T = (n1)/sqrt(2*n^3), which approaches 1/sqrt(2*n) for large n. If we further assume that these two events can be modeled as occurring independently of each other, then the probability that I_n will include both a square and a triangular number approaches P = P_S * P_T = sqrt(1/8)/n. Since sum_{n=1..inf} sqrt(1/8)/n diverges, the sequence should have an infinite number of terms. (The expected number of Ddigit terms would approach log(10)/sqrt(8) = 0.814... as D increases; as it turns out, the average number of Ddigit terms for D in the interval [1,10] is 8/10 = 0.8.)  Jon E. Schoenfield, Oct 09 2014
For a simple algorithm that uses only integer arithmetic, see the Magma program link.  Jon E. Schoenfield, Oct 09 2014


LINKS

Table of n, a(n) for n=1..8.
Jon E. Schoenfield, Magma program


PROG

(C)
#include <stdio.h>
#include <stdlib.h>
typedef unsigned long long U64;
U64 isqrt(U64 a) {
U64 sr = 1ULL<<31;
while (a<sr*sr) sr>>=1;
U64 b = sr>>1;
while (b) {
U64 s = sr + b;
if (a>=s*s) sr = s;
b>>=1;
}
return sr;
}
int main()
{
U64 tn, t, n, cu;
for (t = n = 0; n < 2642246; ++n) {
for (cu = n*n*n; tn <= cu; ++t)
tn += t;
if (tn < cu+n && isqrt(cu) != isqrt(cu+n1))
printf("%llu, ", n);
}
}
(PARI) isok(n) = {st = 0; ss = 0; for (i=n^3+1, n^3+n1, if (issquare(i), ss++); if (ispolygonal(i, 3), st++)); (st == 1) && (ss == 1); } \\ Michel Marcus, Aug 24 2014


CROSSREFS

Cf. A000217, A000290, A002378.
Sequence in context: A183966 A187031 A233642 * A235755 A235530 A116493
Adjacent sequences: A246039 A246040 A246041 * A246043 A246044 A246045


KEYWORD

nonn,hard,more


AUTHOR

Alex Ratushnyak, Aug 23 2014


EXTENSIONS

a(5)a(8) from Hiroaki Yamanouchi, Sep 03 2014


STATUS

approved



