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

1,1

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

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

Example

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.

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]] & ]

Prog

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

Crossrefs

Cf. A000120, A014499.

Sequence in context: A121171 A239790 A065079 * A158201 A350006 A370156

Adjacent sequences: A065046 A065047 A065048 * A065050 A065051 A065052

Keyword

easy,nonn

Author

Robert G. Wilson v, Nov 06 2001

Status

approved