These are exponents such as

160456789 which also sometimes come up in dubious claims or guesses as exponents of Mersenne

primes. In the mersenne.org search space 2<p<999999999, the run of ascending digits is limited to 8 or less and not a multiple of 3. A 3-digit run segment is divisible by 3 always: c c+1 c+2 is c*100 + c*10 + c + 10 + 2 = 111c +12 which will be divisible by 3, since 111=37*3 and 12=4*3.

Both ascending straights and descending straights occur, as well as shuffled straights.

Shuffled straights would be more probable so more common than sorted straights.

Ascending straights seem to me the most interesting of the 3.

I am unaware of any effort to search for or tabulate straights that are prime exponents or the factoring or primality status of the corresponding Mersenne numbers.

There's a

thread about straights as exponents.

One could generalize from

straights to additional valued poker hands.

Top of reference tree:

https://www.mersenneforum.org/showpo...22&postcount=1