Chaitanya's Random Pages

June 26, 2016

2016 has many factors

Filed under: mathematics — ckrao @ 11:16 am

The number 2016 has at least as many factors (36) as any positive integer below it except $1680 = 2^4\times 3\times 5\times 7$ (which has 40 factors). The next time a year will have more factors is $2160 = 2^4\times 3^3\times 5$, also with 40 factors.

Here are the numbers below 2160 also with 36 factors:

• $1260 = 2^2 \times 3^2 \times 5 \times 7$
• $1440 = 2^5 \times 3^2 \times 5$
• $1800 = 2^3 \times 3^2 \times 5^2$
• $1980 = 2^2 \times 3^2 \times 5 \times 11$
• $2016 = 2^5 \times 3^2 \times 7$
• $2100 = 2^2 \times 3 \times 5^2 \times 7$

The first integer with more than 40 factors is $2520 = 2^3 \times 3^2 \times 5 \times 7$ (48 factors).

References

[1] N. J. A. Sloane and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane), A000005 – OEIS.