1,1

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 non-squarefree 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).

R. K. Guy, "Euler's Totient Function", "Solutions of phi(m)=sigma(n)", "Iterations of phi and sigma", "Behavior of phi(sigma(n)) and sigma(phi(n))". =A7 B36-B42 in Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, pp. 138-151, 2004.

Doug Iannucci, in: Gerry Myerson (ed.), 2007 Western Number Theory problems set.

M. F. Hasler, Table of n, a(n) for n=1,...,7499.

(PARI) for( n=1, 10^7, eulerphi(n)==2*eulerphi(sigma(n)) && print1(n", "))

Cf. A137816-A137819, A006872 (phi(sigma(n)) = phi(n)), A067704 (phi(sigma(n)) = 2 phi(n)), A082897.

Sequence in context: A126359 A141408 A107144 this_sequence A089523 A058507 A111057

Adjacent sequences: A137812 A137813 A137814 this_sequence A137816 A137817 A137818

easy,nonn

Richard K. Guy (rkg(AT)cpsc.ucalgary.ca), Richard J. Mathar (mathar(AT)strw.leidenuniv.nl) and M. F. Hasler (MHasler(AT)univ-ag.fr), Feb 11 2008