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

1,2

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.

Formula

Intersection of A002202 and A002191.

Mathematica

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

Prog

(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

Crossrefs

Sequence in context: A090989 A161219 A310664 * A234523 A178549 A244408

Adjacent sequences: A161782 A161783 A161784 * A161786 A161787 A161788

Keyword

nonn

Author

T. D. Noe, Jun 19 2009

Status

approved