Journal of Integer Sequences, Vol. 11 (2008), Article 08.4.2 |

Department of Mathematics, Computer Science and Mechanics

University of Warsaw

Warsaw

Poland

**Abstract:**

The EKG sequence is defined as follows: , and
is the smallest natural number
satisfying
not already in the sequence.
The sequence was previously investigated by Lagarias, Rains and Sloane.
In particular, we know
that is a permutation of the natural numbers and that the
prime numbers appear in this sequence in an increasing order.

Lagarias, Rains and Sloane performed many numerical experiments on the EKG sequence up to the th term and came up with several interesting conjectures. This paper provides proofs for the core part of those conjectures. Namely, let be the sequence with all terms of the form and , for prime, changed to . First, we prove that for any odd prime we have . Then we prove that , i.e., we have except for the values of and for prime: if then , and if then .

(Concerned with sequence A064413 .)

Received March 4 2008;
revised version received September 20 2008.
Published in *Journal of Integer Sequences*, October 2 2008.

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