A137815 Year numbers: numbers n such that phi(n) = 2 phi(sigma(n)).

5, 13, 37, 61, 65, 73, 119, 157, 185, 193, 277, 305, 313, 365, 397, 421, 457, 481, 541, 613, 661, 673, 733, 757, 785, 793, 877, 949, 965, 997, 1093, 1153, 1201, 1213, 1237, 1321, 1381, 1385, 1453, 1547, 1565, 1615, 1621, 1657, 1753, 1873, 1933, 1985, 1993

Offset: 1

Author

R. K. Guy, R. J. Mathar and M. F. Hasler, Feb 11 2008

Keywords

• easy
• nonn

Comments

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).

References

• 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.

Links

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

Crossrefs

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

Programs

Mathematica

Select[Range[2000],EulerPhi[#]==2EulerPhi[DivisorSigma[1,#]]&]  (* Harvey P. Dale, Mar 18 2011 *)

PARI

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