Is there an efficient way to work out how many factors a large number has?

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

Explain why the arithmetic sequence

1, 14, 27, 40, ...

contains many terms of the form 222...2 where only the digit 2 appears. How many twos appear in these terms?