We deduce conditions for additive periodicity of the Sprague᎐Grundy function of Nim-like games, starting with Wythoff's classic game and ending up with a fairly large class of impartial games played on directed graphs. ᮊ 1999 Academic Press

Abstract. An exhaustive search has yielded new sociable groups; one of order 9, two of order 8, and the others of order 4.

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Abstract. For any positive integer n let φ(n) be the Euler function of n. A positive integer n is called a noncototient if the equation x − φ(x) = n has no solution x. In this note, we give a sufficient condition on a positive integer k such that the geometrical progression (2 m k) m≥1 consists entirely of noncototients. We then use computations to detect seven such positive integers k.