A227219 Smallest triangular number ending in at least n 5's.

15, 55, 6555, 2235555, 717655555, 13140555555, 9206385555555, 1551745755555555, 103835341555555555, 26427628585555555555, 2262637355455555555555, 8808604932555555555555, 4704913988655555555555555, 4704913988655555555555555, 19391196055786555555555555555

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(3)=6555 because 6555 is the smallest triangular number ending in 3 '5's.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..501

Crossrefs

Cf. A000217, A187127, A227218, A227220.

Programs

Mathematica

t = {}; Do[Do[x = n*(n + 1)/2;If[Mod[x, 10^m] == 5*(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 + 5*(10^n - 1)/9 && 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 A227219(n):
    k, a = 10**n<<1, 10*(10**n-1)//9
    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(1) corrected and a(11)-a(15) from Giovanni Resta, Sep 20 2013

Name clarified by Michel Marcus, Sep 05 2019