A214588 Primes p such that p mod 16 < 8.

2, 3, 5, 7, 17, 19, 23, 37, 53, 67, 71, 83, 97, 101, 103, 113, 131, 149, 151, 163, 167, 179, 181, 193, 197, 199, 211, 227, 229, 241, 257, 263, 277, 293, 307, 311, 337, 353, 359, 373, 389, 401, 419, 421, 433, 439, 449, 467, 487, 499, 503, 547, 563, 577, 593, 599

Offset

1,1

Comments

Original definition: Primes p such that p XOR 8 = p + 8.

This is an example of a class of primes p such that p XOR n = p + n.

A002144 is the case where n=2, there are no cases where n=3, in A033203 n=4, 2 is the only p for n=5, in A007519 n=6, there are no cases where n=7. This sequence occurs when n=8.

In general if n is an odd number in A004767 there are no primes, if n is an odd number in A016813, then 2 is the only prime, and if n is an even number (A005843) there is a set of primes that satisfies the relationship p XOR n = p + n.

Links

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Example

103 is in the sequence because 103 mod 16 is 7 which is less than 8. - Indranil Ghosh, Jan 18 2017

Mathematica

Select[Prime[Range[200]], Mod[#, 16]<8&] (* Harvey P. Dale, Jan 11 2018 *)

Prog

(Magma)
XOR := func<a, b | Seqint([ (adigs[i] + bdigs[i]) mod 2 : i in [1..n]], 2)
                       where adigs := Intseq(a, 2, n)
                       where bdigs := Intseq(b, 2, n)
                       where n := 1 + Ilog2(Max([a, b, 1]))>;
for n in [2 .. 1000] do
   if IsPrime(n)  then  pn:=n;
      if (XOR(pn, 8) eq pn+8) then pn; end if;
   end if;
end for;
(PARI) is_A214588(p)={ !bittest(p, 3) & isprime(p) } \\ M. F. Hasler, Jul 24 2012
(PARI) forprime(p=1, 699, bittest(p, 3) || print1(p", ")) \\ M. F. Hasler, Jul 24 2012
(Python)
from sympy import isprime
i=1
j=1
while j<=10000:
....if  isprime(i)==True and (i%16)<8:
........print str(j)+" "+str(i)
........j+=1
....i+=1 # Indranil Ghosh, Jan 18 2017

Crossrefs

Cf. A022144, A033203, A007519, A004767, A016813, A005843.

Sequence in context: A108222 A090725 A276141 * A089968 A164060 A356405

Adjacent sequences: A214585 A214586 A214587 * A214589 A214590 A214591

Keyword

nonn

Author

Brad Clardy, Jul 22 2012

Status

approved