A065049 - OEIS

%I #20 Apr 03 2023 10:36:10

%S 11,41,43,47,59,107,131,137,139,163,167,173,179,191,227,233,239,251,

%T 277,337,349,373,419,431,443,491,521,523,547,557,563,569,571,587,617,

%U 619,641,643,647,653,659,673,677,691,701,719,739,743,751,761,809,811

%N Odd primes of incorrect parity: number of 1's in the binary representation of n (mod 2) == 1 - (n mod 3) (mod 2). Also called isolated primes.

%C "The prime maze - consider the prime numbers in base 2, starting with the smallest prime (10)2. One can move to another prime number by either changing only one digit of the number, or adding a 1 to the front of the number. Can we reach 11 = (1011)2.? 333? The Mersennes?" - Caldwell

%H Robert Israel, <a href="/A065049/b065049.txt">Table of n, a(n) for n = 1..10000</a>

%H Chris K. Caldwell, <a href="https://t5k.org/links/curiosities/problems_and_puzzles/">Prime Links + +</a>

%H W. Paulsen, <a href="http://www.csm.astate.edu/~wpaulsen/primemaze/pmaze.html">The Prime Maze</a>, Fib. Quart., 40 (2002), 272-279.

%H C. Rivera, <a href="http://www.primepuzzles.net/problems/prob_025.htm">Problem 25 - William Paulsen's Prime Numbers Maze</a>

%e 47 is in the sequence because 47d = 101111b which has five 1's in its binary notation; an odd number. Also 47 == 2 (mod 3); an even number. Therefore a mismatch exists.

%p filter:= proc(n) convert(convert(n,base,2),`+`) + (n mod 3) mod 2 = 1 end proc:

%p select(filter, [seq(ithprime(i),i=2..1000)]); # _Robert Israel_, Jun 19 2018

%t Select[ Range[3, 1000, 2], PrimeQ[ # ] && EvenQ[ Count[ IntegerDigits[ #, 2], 1]] == OddQ[ Mod[ #, 3]] & ]

%o (PARI) isok(p) = (p>2) && isprime(p) && ((hammingweight(p) % 2) != ((p % 3) % 2)); \\ _Michel Marcus_, Dec 15 2018

%Y Cf. A000120, A014499.

%K easy,nonn

%O 1,1

%A _Robert G. Wilson v_, Nov 06 2001