A binomial number is a number of the form , where , and are integers. Binomial numbers
can be factored algebraically as

(1)

for all ,

(2)

for
odd, and

(3)

for all positive integers . For example,

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

and

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

Rather surprisingly, the number of factors of with and symbolic and a positive integer is given by , where is the number of divisors of and is the divisor function. The first few terms are
therefore 1, 2, 2, 3, 2, 4, 2, ... (OEIS A000005).

Similarly, the number of factors of is given by , where is the number of odd divisors of
and is the odd
divisor function. The first few terms are therefore 1, 1, 2, 1, 2, 2, 2, 1,...
(OEIS A001227).

In 1770, Euler proved that if , then every odd factor of

Guy, R. K. "When Does Divide ." §B47 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 102,
1994.Qi, S and Ming-Zhi, Z. "Pairs where Divides for All ." Proc. Amer. Math. Soc.93, 218-220, 1985.Schinzel,
A. "On Primitive Prime Factors of ." Proc. Cambridge Phil. Soc.58,
555-562, 1962.Sloane, N. J. A. Sequences A000005/M0246
and A001227 in "The On-Line Encyclopedia
of Integer Sequences."