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Following D. Iannucci, n is called a "year number" if phi(n) / phi(sigma(n)) = 2 (thus 365 is a year number, explaining the terminology).

D. Iannucci asks: Are there any even year numbers? Are there any odd year numbers that are not squarefree?

Remark: If n = q_1 q_2 ... q_k is a product of odd primes such that (q_j + 1)/2 is an odd prime for all j, then n is a year number.

Solution: for nonsquarefree year numbers, see A137816. See A137817-A137819 for year numbers with cubes, 4th powers, 5th powers.

Eric Landquist found year numbers divisible by 7^2, 7^3 and 7^4, as well as 120781449 = 3^8 * 41 * 449.

The existence of even year numbers is still open, but Eric checked all 200-smooth even integers with a single large prime up to 10^8 and found no year numbers among them.

See also references in A082897 (perfect totient numbers).

This is the subsequence of nonsquarefree elements of A137815. See there for more comments and references.

There are only 32 such numbers below 10^6, 145 below 10^7 and 785 below 10^8.

See A137815 for general comments and references about "year numbers". This is the subsequence of cubeful elements of A137815, i.e. its intersection with A046099. As such, it is of course also a subsequence of A137816.

There are only 56 such numbers below 10^8.

See A137815 for general comments and references about "year numbers". This is the subsequence of elements of A137815 divisible by a biquadrateful number, i.e. its intersection with A046101 (numbers divisible by the 4th power of some prime). As such, it is of course also a subsequence of A137817 and a fortiori of A137816.

There are only 28 such numbers below 10^8.