A227220 Smallest triangular number ending in n 6's.

6, 66, 666, 8146666, 2340066666, 109704666666, 15212926666666, 66488766666666, 173147535666666666, 11302559226666666666, 14642337066666666666, 77850029701666666666666, 1029964845546666666666666, 3086210323568466666666666666, 4014564887872666666666666666

Offset: 1

Author

Shyam Sunder Gupta, Sep 19 2013

Keywords

• nonn
• base

Comments

If a triangular number ends in like digits then it can only end in 0's, 1's, 5's or 6's.

Examples

			a(4)=8146666 because 8146666 is the smallest triangular number ending in  4 '6's.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A000217, A187127, A227218, A227219.

Programs

Maple

f:= proc(n) local k;
   k:=min(map(rhs@op, [msolve(k*(k+1)=4/3*(10^n-1),2*10^n)]));
   k*(k+1)/2
end proc:
map(f, [$1..30]); # Robert Israel, Sep 04 2019

Mathematica

t = {}; Do[Do[x = n*(n + 1)/2;If[Mod[x, 10^m] == 6*(10^m - 1)/9, AppendTo[t, x]; Break[]], {n, 1, 10^m}], {m, 1, 10}]; t
a[n_] := Block[{x,y,s}, s = y /. List@ ToRules[ Reduce[(y+1)* y/2 == x*10^n + 2(10^n - 1)/3 && y > 0 && x >= 0, {y, x}, Integers] /. C[1] -> 0]; Min[s*(s + 1)/2]]; Array[a,20] (* Giovanni Resta, Sep 20 2013 *)

Python

from sympy import sqrt_mod_iter
def A227220(n):
    k, a = 10**n<<1, (10**n-1)//3<<2
    m = (a<<2)+1
    return min(b for b in ((d>>1)*((d>>1)+1) for d in sqrt_mod_iter(m, k) if d&1) if b%k==a)>>1 # Chai Wah Wu, May 04 2024

Extensions

a(11)-a(15) from Giovanni Resta, Sep 20 2013