Concatenation of two Mersenne numbers

[enzocreti]

Concatenate 2^k-1 and 2^(k-2)-1. Example 71, 153,...can you find a prime congruent to 4 or 3 mod 7?

[Dr Sardonicus]

[Stargate38]

Here's the remainders of the first few numbers of that form:

71 mod 7 = 1
153 mod 7 = 6
317 mod 7 = 2
6315 mod 7 = 1
12731 mod 7 = 5
25563 mod 7 = 6
511127 mod 7 = 1
1023255 mod 7 = 2
2047511 mod 7 = 4
40951023 mod 7 = 1
81912047 mod 7 = 0
163834095 mod 7 = 5
327678191 mod 7 = 1
6553516383 mod 7 = 1
13107132767 mod 7 = 1
26214365535 mod 7 = 1
524287131071 mod 7 = 4

Out of those 17 numbers, only 71 and 317 are prime. It seems that the only possible remainders are 0, 1, 2, 4, 5, and 6.

[enzocreti]

the possible remainders are 0,1,2,3,4,5,6 (3 with frequency 1/18)
Can you find a prime with residue 3 or 4?

[henryzz]

PRPs:

M(3)M(1) 1
M(5)M(3) 2
M(25)M(23) 6
M(33)M(31) 1
M(143)M(141) 2
M(165)M(163) 1
M(285)M(283) 1
M(321)M(319) 1
M(323)M(321) 2
M(1003)M(1001) 5
M(1683)M(1681) 1
M(1921)M(1919) 5
M(4223)M(4221) 1
M(5451)M(5449) 1
M(6783)M(6781) 1

The number following M(x)M(x-2) is the number mod 7