Prime Tau Function

The Prime Tau function of n is the total number of positive divisors of n.

To compute Tau(n), first factor n into its prime factors. For this document the following notation is used. An integral number n > 1 can be written as

where the Pi's are the various different prime factors, Ai the number of times Pi occurs in the prime factorization and r the number of prime factors.

The formula for Tau is:

For example, n = 100 = 2^2 * 5^2
Tau(100) = (2+1)*(2+1)= 9.
They are:
2^0 * 5^0 = 1,
2^1 * 5^0 = 2,
2^2 * 5^0 = 4,
2^0 * 5^1 = 5,
2^1 * 5^1 = 10,
2^2 * 5^1 = 20,
2^0 * 5^2 = 25,
2^1 * 5^2 = 50,
2^2 * 5^2 = 100.

For n = 67500 = 2^2 * 3^3 * 5^4
Tau(67500) = (2+1)*(3+1)*(4+1) = 3*4*5 = 60.

And: Divisor -- From MathWorld

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This page accessed times since March 26, 2005.
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Changes last made on Monday, 06-Aug-07 20:47:30 PDT

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