A359664 Prime Maze Room 11, opposite parity of A059459 starting from prime room 11.

11, 43, 41, 2089, 2081, 2083, 2087, 10889035741470030830827987437816582768679, 10889035741470030830827987437816582768647

Offset: 1

Author

Gregory Allen, Jan 10 2023

Keywords

• nonn
• less

Comments

This is the opposite parity sequence of A059459 and lexicographically least of this sequence.

It is currently not known whether both of these sequences are infinite.

I was able to calculate 40 terms; a(40) is a 3261-digit prime.

a(1) = 11; a(n+1) is obtained by writing a(n) in binary and trying to complement just one bit, starting with the least significant bit, until a new prime is reached. (Terms 2 and 3 are excluded values from the main sequence.)

Conjecture: Room 2 and Room 11 are unlinked, i.e., two separate mazes or branches/trees, as they are of opposite parities.

Links

• Gregory Allen, Table of n, a(n) for n = 1..40
• William Paulsen, Prime Maze, See Prime room 11.
• Carlos Rivera, Problem 25. William Paulsen's Prime Numbers Maze, The Prime Puzzles and Problems Connection.

Crossrefs

Cf. A059459.

Programs

Mathematica

maxBits = 2^14;
ClearAll[a];
a[1] = 3;
a[2] = 2;
a[3] = 11;
n = 4;
a[n_] :=
 a[n] = If[PrimeQ[a[n - 1]],
   bits = PadLeft[IntegerDigits[a[n - 1], 2], maxBits];
   For[i = 1, i <= maxBits, i++, bits2 = bits;
    bits2[[-i]] = 1 - bits[[-i]];
    If[i == maxBits, Print["maxBits reached"]; Break[],
     If[PrimeQ[an = FromDigits[bits2, 2]] &&
       FreeQ[Table[a[k], {k, 1, n - 1}], an], Return[an]]]],
   0]; Table[a[n], {n, 42}]