By Alfred North Whitehead
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Elem. Math. 14 (1959), 60–61. [6] Y. Wang, On the representation of large integer as a sum of prime and an almost prime. Sci. Sinica 11 (1962), 1033–1054. Originally published in Colloquium Mathematicum LXVIII (1995), 55–58 Andrzej Schinzel Selecta On integers not of the form n − ϕ(n) with J. Browkin (Warszawa) W. Sierpi´nski asked in 1959 (see [4], pp. 200–201, cf. [2]) whether there exist infinitely many positive integers not of the form n − ϕ(n), where ϕ is the Euler function. We answer this question in the affirmative by proving Theorem.
Erd˝os’ paper [1]. Using ideas and results from that paper we can prove the following theorem. Theorem 3. Let f (n) be an additive function satisfying condition 1 of Theorem 1 and let (1/p) be divergent, f (p) /p convergent, then the distribution function of f (p)=0 h-tuples f (m + 1), f (m + 2), . . , f (m + h) exists, and it is a continuous function. Proof. We denote by N (f ; c1 , c2 , . . , ch ) the number of positive integers m not exceeding n for which f (m + i) ci , i = 1, 2, . . , h, where ci are given constants.
P|m 888 G. Arithmetic functions Let us also consider the function fk (m) = f (p). We are going to show that p|m, p k the sequence N (fk ; c1 , c2 , . . , ch )/n is convergent. Since fk (m + A) = fk (m), where A= p, we can see that the integers m for which p k c fk (m + i) c ci (i = 1, 2, . . , h) are distributed periodically with the period A. Hence N (fk ; c1 , c2 , . . , ch )/n has a limit. To prove the existence of a limit of N (f ; c1 , c2 , . . , ch )/n it is sufficient to show that for arbitrary ε > 0 there exists k0 such that for every k > k0 and n > n(ε) N (f ; c1 , c2 , .