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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.
6
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
Chris K. Caldwell, Prime Links + +
W. Paulsen, The Prime Maze, Fib. Quart., 40 (2002), 272-279.
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
Sequence in context: A121171 A239790 A065079 * A158201 A350006 A370156
KEYWORD
easy,nonn
AUTHOR
Robert G. Wilson v, Nov 06 2001
STATUS
approved