A065049 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.

11, 41, 43, 47, 59, 107, 131, 137, 139, 163, 167, 173, 179, 191, 227, 233, 239, 251, 277, 337, 349, 373, 419, 431, 443, 491, 521, 523, 547, 557, 563, 569, 571, 587, 617, 619, 641, 643, 647, 653, 659, 673, 677, 691, 701, 719, 739, 743, 751, 761, 809, 811

Offset: 1

Author

Robert G. Wilson v, Nov 06 2001

Keywords

• easy
• nonn

Comments

"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

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Chris K. Caldwell, Prime Links + +
• W. Paulsen, The Prime Maze, Fib. Quart., 40 (2002), 272-279.
• C. Rivera, Problem 25 - William Paulsen's Prime Numbers Maze

Crossrefs

Cf. A000120, A014499.

Programs

Maple

filter:= proc(n) convert(convert(n,base,2),`+`) + (n mod 3) mod 2 = 1 end proc:
select(filter, [seq(ithprime(i),i=2..1000)]); # Robert Israel, Jun 19 2018

Mathematica

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

PARI

isok(p) = (p>2) && isprime(p) && ((hammingweight(p) % 2) != ((p % 3) % 2)); \\ Michel Marcus, Dec 15 2018