%I #5 Nov 17 2022 07:20:26

%S 3343,42293,461393,70793,72053,268267,8917219

%N Consider the graph of symmetric primes where p and q are connected if |p-q| = gcd(p-1,q-1). This sequence is an irregular table where the n-th row lists the first symmetric prime in a connected component with n vertices, with one representative for each nonisomorphic graph. Within a row, graphs are ordered by increasing size of its initial prime.

%C Row lengths are A001349(n); if the sequence is finite the last row may be shorter.

%C Kalmynin gives T(2, 1) = 3343 and proves that, under a conjecture which is intermediate between Dickson's conjecture and the Bateman-Horn-Stemmler conjecture, that this sequence is infinite.

%H A. B. Kalmynin, <a href="http://math.colgate.edu/~integers/v2/v2.pdf">On the Symmetry Graph of Prime Numbers</a>, INTEGERS 21 (2021), #A2.

%e T(2, 1) = 3343 has components {3343, 4457} which form the complete graph K_2.

%e T(3, 1) = 42293 has components {42293, 42487, 63439} which form the path graph P_3.

%e T(3, 2) = 461393 has components {461393, 519067, 692089} which form the complete graph K_3.

%e T(4, 1) = 70793 has components {70793, 106187, 106189, 123887} which form the claw graph.

%e T(4, 2) = 72053 has components {72053, 108079, 216157, 288209} which form the path graph P_4.

%e T(4, 3) = 268267 has components {268267, 357689, 536531, 536533} which form the paw graph.

%e T(4, 4) = 8917219 has components {8917219, 9908021, 14862031, 17834437} which form the square graph.

%Y Cf. A090190.

%K nonn,tabf,hard,more

%O 2,1

%A _Charles R Greathouse IV_, Nov 08 2022