Joint coprimality ($\gcd (a_1,\dots ,a_r)=1$)

A list of integers $a_1,\dots ,a_r$, with $r\ge 2$, has joint coprimality when its greatest common divisor is one; define that greatest common divisor by repeated application of the two-input operation. For positive integers, no prime factor may divide every entry. Thus $6,10,15$ have joint coprimality but fail pairwise coprimality. For two entries, the conditions coincide.