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Let's say that a positive integer $n$ is a happy mother if $6$ divides $n$ and $(n-1,n+1)$ is a pair of twin primes. Is the difference between two consecutive happy mothers necessarily a happy mother itself?
Thanks in advance.

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up vote 4 down vote accepted

No; the numbers $312$ and $348$ are consecutive happy mothers, but their difference $36$ is not.

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No, $30$ and $6$ are happy mothers, but $24$ is not.

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But $6$ and $30$ are not consecutive happy mothers, there are $12$ and $18$ between them. – Sylvain Julien May 17 '14 at 12:17

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