A365001 Primes from which it is not possible to reach a (different) Mersenne prime by toggling a single bit per step while still remaining prime at every step.

73, 89, 127, 173, 191, 233, 239, 251, 257, 277, 337, 349, 373, 431, 443, 491, 557, 653, 683, 701, 733, 761, 769, 773, 787, 853, 907, 911, 971, 1019, 1093, 1109, 1117, 1193, 1201, 1237, 1297, 1301, 1303, 1361, 1367, 1373, 1381, 1399, 1429, 1453, 1489, 1493

Offset: 1

Author

Sean A. Irvine, Aug 15 2023

Keywords

• nonn
• base

Comments

These are "locations" in The Prime Number Maze from which it is not possible to reach a (different) Mersenne prime by successively toggling single bits (see Paulsen for exact rules). This differs from A065111 in that it contains locations, such as 2131099, which are not reachable from 2.

Examples

			For 73 the only available move is to swap to 89, and vice versa (although there are other ways of reaching them, for example 601 can transition to 89). While 127 is already a Mersenne prime, it is not possible to reach another Mersenne prime starting from 127.
		

Links

• Sean A. Irvine, Java program (github)
• W. Paulsen, The Prime Maze
• W. Paulsen, Are some rooms totally isolated?
• W. Paulsen, The Prime Maze, Fib. Quart., 40 (2002), 272-279.
• Carlos Rivera, Problem 25. The William Paulsen's Prime Numbers Maze, The Prime Puzzles & Problems Connection.

Crossrefs

Cf. A065111 (reachable from 2), A065092 (singularly dead end primes).

Extensions

Missing terms inserted by Andrew Howroyd and name clarified by Sean A. Irvine, Sep 21 2023