login
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

%I #27 Aug 17 2017 11:57:47

%S 1,6,45,-1,15,36,21,120,-1,210,66,36,78,210,45,-1,1326,378,190,120,

%T 210,66,1035,4560,-1,78,1485,1540,435,300,465,902496,1485,3570,105,-1,

%U 17205,4560,2145,120,861,630,903,157080,4095,276,9870,41328,-1,300,153,780,5565,1185030

%N 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.

%C 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

%H Robert Israel, <a href="/A077672/b077672.txt">Table of n, a(n) for n = 1..10000</a>

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

%p f:= proc(n) local eq, X,Y,S,i,Si,XY,y;

%p if issqr(n) then return -1 fi;

%p eq:= n*(X^2-1)=Y^2-1;

%p S:= map(t -> subs(t, [X,Y]), [isolve(eq)]);

%p for i from 0 do

%p Si:= select(t -> t[1] > 3 and t[1]::odd and t[2]>0, expand(subs(_Z1=i, S)));

%p if Si <> [] then

%p XY:= Si[min[index](map(t -> t[1],Si))];

%p y:= (XY[2]-1)/2;

%p return y*(y+1)/2;

%p fi

%p od

%p end proc:

%p f(1):= 1:

%p map(f, [$1..100]); # _Robert Israel_, Aug 15 2017

%t 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 *)

%o (PARI) isok(k, n) = {my(t = k*(k+1)/2); !(t % n) && (t/n != 1) && ispolygonal(t/n, 3);}

%o 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

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

%K sign

%O 1,2

%A _Amarnath Murthy_, Nov 16 2002

%E More terms from _Sascha Kurz_, Jan 27 2003

%E More terms from _Michel Marcus_, Aug 15 2017