A358312 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.

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

Offset: 2

Author

Charles R Greathouse IV, Nov 08 2022

Keywords

• nonn
• tabf
• hard
• more

Comments

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

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.

Examples

			T(2, 1) = 3343 has components {3343, 4457} which form the complete graph K_2.
T(3, 1) = 42293 has components {42293, 42487, 63439} which form the path graph P_3.
T(3, 2) = 461393 has components {461393, 519067, 692089} which form the complete graph K_3.
T(4, 1) = 70793 has components {70793, 106187, 106189, 123887} which form the claw graph.
T(4, 2) = 72053 has components {72053, 108079, 216157, 288209} which form the path graph P_4.
T(4, 3) = 268267 has components {268267, 357689, 536531, 536533} which form the paw graph.
T(4, 4) = 8917219 has components {8917219, 9908021, 14862031, 17834437} which form the square graph.
		

Links

• A. B. Kalmynin, On the Symmetry Graph of Prime Numbers, INTEGERS 21 (2021), #A2.

Crossrefs

Cf. A090190.