A161785 Numbers k that are in the range of both Euler's phi function and the sigma function.

1, 4, 6, 8, 12, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 54, 56, 60, 72, 78, 80, 84, 96, 102, 104, 108, 110, 112, 120, 126, 128, 132, 138, 140, 144, 150, 156, 160, 162, 164, 168, 176, 180, 192, 198, 200, 204, 210, 212, 216, 222, 224, 228, 240, 252, 256, 260

Offset: 1

Author

T. D. Noe, Jun 19 2009

Keywords

• nonn

Comments

That is, for each k there exist x and y such that k = phi(x) = sigma(y). Sigma is the sum of divisors function. Ford, Luca, and Pomerance prove that this sequence is infinite.

References

• R. K. Guy, Unsolved Problems in Number Theory, B38.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Kevin Ford, Florian Luca, and Carl Pomerance, Common values of the arithmetic functions phi and sigma, Bull. London Math. Soc. 42 (2010), pp. 478-488.

Programs

Mathematica

Intersection[EulerPhi[Range[9660]], DivisorSigma[1,Range[2112]]]

PARI

list(lim)={
   my(u=vector(lim\=1,k,sigma(k)),v=vector(if(lim>63,3*lim*log(log(lim))\1,210),k,eulerphi(k)));
    select(n->n<=lim,setintersect(vecsort(v,,8),vecsort(u,,8)))
}; \\ Charles R Greathouse IV, Feb 05 2013

Formula

Intersection of A002202 and A002191.