Computer experiments show that singly even multiples of 6 sur-rounded by prime pairs exhibit a larger ratio of nonsquarefree to squarefree multiples than generic singly even multiples of 6, a bias of ca 10.6% measured against the expected value. The same bias occurs for isolated primes next to singly even multiples of 6; here the deviation from the expected value is ca 3.3% of this value. The expected value of the ratio of singly even to doubly even nonsquarefree multiples of 6 also differs from values found experimentally for prime pairs centered on such multiples or isolated primes next to them. For pairs, this ratio exceeds its unbiased value by ca 6.2%, for isolated primes by ca 2.0%. The values cited are for the first 10^10 primes, the largest range we investigated. This paper broadens our recent study of a newly found bias in the distribution of primes by examining singly and doubly even multiples of 6. In particular, it shows that for primes centered on or next to singly even multiples of 6, the statistical biases in question are more pronounced than in the general case studied by us before.