A077672 a(1) = 1; for n > 1, a(n) = smallest triangular number which is n times another triangular number > 1, or -1 if no such number exists.

1, 6, 45, -1, 15, 36, 21, 120, -1, 210, 66, 36, 78, 210, 45, -1, 1326, 378, 190, 120, 210, 66, 1035, 4560, -1, 78, 1485, 1540, 435, 300, 465, 902496, 1485, 3570, 105, -1, 17205, 4560, 2145, 120, 861, 630, 903, 157080, 4095, 276, 9870, 41328, -1, 300, 153, 780, 5565, 1185030

Offset

1,2

Comments

a(b^2) = -1 for b > 1 because (2*b*m + b - 1)^2 < 1 + 4*b^2*m^2 + 4*b^2*m < (2*b*m + b)^2. - Sascha Kurz, Jan 27 2003

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(5) = 15 = 5*3, a(6) = 630 = 105*6.

Maple

f:= proc(n) local eq, X, Y, S, i, Si, XY, y;
  if issqr(n) then return -1 fi;
  eq:= n*(X^2-1)=Y^2-1;
  S:= map(t -> subs(t, [X, Y]), [isolve(eq)]);
  for i from 0 do
     Si:= select(t -> t[1] > 3 and t[1]::odd and t[2]>0, expand(subs(_Z1=i, S)));
     if Si <> [] then
       XY:= Si[min[index](map(t -> t[1], Si))];
       y:= (XY[2]-1)/2;
       return y*(y+1)/2;
     fi
  od
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Aug 15 2017

Mathematica

With[{s = Map[# (# + 1)/2 &, Range[10^4]]}, Table[2 Boole[n == 1] + If[IntegerQ[Sqrt@ n], -1, SelectFirst[n Rest@ s, MemberQ[s, #] &]], {n, 54}]] (* Michael De Vlieger, Aug 15 2017 *)

Prog

(PARI) isok(k, n) = {my(t = k*(k+1)/2); !(t % n) && (t/n != 1) && ispolygonal(t/n, 3); }
a(n) = {if (n == 1, return (1)); if (issquare(n), return (-1)); my(k = 1); while (!isok(k, n), k++); k*(k+1)/2; } \\ Michel Marcus, Aug 15 2017

Crossrefs

Cf. A000217 (triangular numbers), A000290 (squares).

Sequence in context: A271963 A065783 A158460 * A119202 A367561 A286325

Adjacent sequences: A077669 A077670 A077671 * A077673 A077674 A077675

Keyword

sign

Author

Amarnath Murthy, Nov 16 2002

Extensions

More terms from Sascha Kurz, Jan 27 2003

More terms from Michel Marcus, Aug 15 2017

Status

approved