%I #14 Aug 09 2015 15:04:05
%S 5,13,37,61,65,73,119,157,185,193,277,305,313,365,397,421,457,481,541,
%T 613,661,673,733,757,785,793,877,949,965,997,1093,1153,1201,1213,1237,
%U 1321,1381,1385,1453,1547,1565,1615,1621,1657,1753,1873,1933,1985,1993
%N Year numbers: numbers n such that phi(n) = 2 phi(sigma(n)).
%C 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).
%C D. Iannucci asks: Are there any even year numbers? Are there any odd year numbers that are not squarefree?
%C 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.
%C Solution: for nonsquarefree year numbers, see A137816. See A137817-A137819 for year numbers with cubes, 4th powers, 5th powers.
%C Eric Landquist found year numbers divisible by 7^2, 7^3 and 7^4, as well as 120781449 = 3^8 * 41 * 449.
%C 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.
%C See also references in A082897 (perfect totient numbers).
%D 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.
%D Doug Iannucci, in: Gerry Myerson (ed.), 2007 Western Number Theory problems set.
%H M. F. Hasler, <a href="/A137815/b137815.txt">Table of n, a(n) for n=1,...,7499</a>.
%t Select[Range[2000],EulerPhi[#]==2EulerPhi[DivisorSigma[1,#]]&] (* _Harvey P. Dale_, Mar 18 2011 *)
%o (PARI) for( n=1,10^7, eulerphi(n)==2*eulerphi(sigma(n)) && print1(n", "))
%Y Cf. A137816-A137819, A006872 (phi(sigma(n)) = phi(n)), A067704 (phi(sigma(n)) = 2 phi(n)), A082897.
%K easy,nonn
%O 1,1
%A _R. K. Guy_, _R. J. Mathar_ and _M. F. Hasler_, Feb 11 2008