triangular number counting function

For a given nonnegative number $x$ , the triangular number counting function counts how many triangular numbers are not greater than $x$ . The formula is simple:

\lfloor\frac{-1+\sqrt{8x+1}}{2}\rfloor,

in sharp contrast to the lack of a formula for the prime counting function $\pi(x)$ . If one accepts 0 as a triangular number, the formula easily accommodates this by the mere addition of 1 after flooring the fraction.

References

1 Zhi-Wei Sun, “On Sums of Primes and Triangular Numbers” ArXiv preprint, 10 April (2008): 1

Title	triangular number counting function
Canonical name	TriangularNumberCountingFunction
Date of creation	2013-03-22 18:03:03
Last modified on	2013-03-22 18:03:03
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	5
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11A25