An Extension of the Euler Phi-function to Sets of Integers Relatively Prime to 30

摘要

Let $n geq 1$ be an integer and let $S= {1,7,11,13,17,19,23,29},$ the set of integers which are both less than and relatively prime to $30.$ Define $phi_3(n)$ to be the number of integers $x, ; 0 leq x leq n-1,$ for which $gcd(30n, 30x+i) = 1$ for all $i in S.$ In this note we show that? $phi_3$ is multiplicative, that is, if $gcd(m, n)=1,$ then $phi_3(mn)=phi_3(m)phi_3(n).$ We make a conjecture about primes generated by S.