I was messing around to see how high N could get before going over 10 to the power of Y where N is the sum of all previous numbers from 1 to N. For example, starting with 10¹ or 10, N would be 4 (1 2 3 4) and the sum of N would be 10.

What I noticed is that N oscillates back and forth between two ranges dependent upon whether 10 is raised to an even power or odd power.

Below is some data with a simple Python script to illustrate the concept.

Thanks!

#!/usr/bin/env python3

import time

sum = 0
for y in range (1,16):
    n = 0
    while sum <= 10**y:
        n  = 1
        sum  = n
    print(n-1, sum-n)
    sum = 0

And here is some data showing the oscillation:

N | N'th Triangle Number
4 10  (Y = 10 or 10^1)
13 91 (Y = 100 or 10^2) etc.
44 990 
140 9870
446 99681
1413 998991
4471 9997156
14141 99991011
44720 999961560
141420 9999878910
447213 99999957291
1414213 999999911791
4472135 9999997965180

Why is N oscillating between 4xxxxxx and 14xxxxx ?