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

1,1

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.

Links

Table of n, a(n) for n=1..48.

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.

Example

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.

Crossrefs

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

Sequence in context: A033247 A014754 A007766 * A065111 A325070 A152308

Adjacent sequences: A364998 A364999 A365000 * A365002 A365003 A365004

Keyword

nonn,base

Author

Sean A. Irvine, Aug 15 2023

Extensions

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

Status

approved