While studying some $k-$tuples of primes I found the following $\mathcal{H}=\{0,2,6,12,20,26,30,32\}$ which seems admissible to me.
using the primes package in Rstudio I get the following $p_1$ between $1$ and $100000000$ :
$348431, 26074901, 32624981, 43713557, 51877097, 64981067, 67787537, 73184621, 74904101, 80372681$
$p_1$ primes appear such that $p_1\equiv 1\pmod{10}$ and $p_1\equiv 7\pmod{10}$
According to my calculations, each $p_1\equiv 1\pmod{10}$ should be $348431\pmod{3484470}$.
and each $p_1\equiv 7\pmod{10}$ should be $43713557\pmod{437135730}$.
I see that they are not in the arithmetic progression $3484470n+348431$ or $437135730n+43713557$.
Could someone tell me what I'm doing wrong, in what arithmetic progression can I "put" these primes?
(I must add that it seems strange to me that this sequence of primes does not appear in OEIS, this makes me more doubtful about it)
This collection is not admissable since we can add $8$ into the list and still have a collection (probably) giving infinite many prime tuples.
The smallest number (after $11$) giving a prime-tuple is $$29597411$$ Every such number must be of the form $210k+11$. Not sure whether we can insert further even numbers without destroying this property.