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A214588 Primes p such that p mod 16 < 8.

%I A214588 #26 Jan 09 2025 15:43:32
%S A214588 2,3,5,7,17,19,23,37,53,67,71,83,97,101,103,113,131,149,151,163,167,
%T A214588 179,181,193,197,199,211,227,229,241,257,263,277,293,307,311,337,353,
%U A214588 359,373,389,401,419,421,433,439,449,467,487,499,503,547,563,577,593,599
%N A214588 Primes p such that p mod 16 < 8.
%C A214588 Original definition: Primes p such that p XOR 8 = p + 8.
%C A214588 This is an example of a class of primes p such that p XOR n = p + n.
%C A214588 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.
%C A214588 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.
%H A214588 Indranil Ghosh, <a href="/A214588/b214588.txt">Table of n, a(n) for n = 1..10000</a>
%e A214588 103 is in the sequence because 103 mod 16 is 7 which is less than 8. - _Indranil Ghosh_, Jan 18 2017
%t A214588 Select[Prime[Range[200]],Mod[#,16]<8&] (* _Harvey P. Dale_, Jan 11 2018 *)
%o A214588 (Magma)
%o A214588 XOR := func<a, b | Seqint([ (adigs[i] + bdigs[i]) mod 2 : i in [1..n]], 2)
%o A214588                        where adigs := Intseq(a, 2, n)
%o A214588                        where bdigs := Intseq(b, 2, n)
%o A214588                        where n := 1 + Ilog2(Max([a, b, 1]))>;
%o A214588 for n in [2 .. 1000] do
%o A214588    if IsPrime(n)  then  pn:=n;
%o A214588       if (XOR(pn,8) eq pn+8) then pn; end if;
%o A214588    end if;
%o A214588 end for;
%o A214588 (PARI) is_A214588(p)={ !bittest(p,3) & isprime(p) } \\ _M. F. Hasler_, Jul 24 2012
%o A214588 (PARI) forprime(p=1,699, bittest(p,3) || print1(p",")) \\ _M. F. Hasler_, Jul 24 2012
%o A214588 (Python)
%o A214588 from sympy import isprime
%o A214588 i=1
%o A214588 while i<=600:
%o A214588     if  isprime(i)==True and (i%16)<8:
%o A214588         print(i, end=", ")
%o A214588     i+=1 # _Indranil Ghosh_, Jan 18 2017
%Y A214588 Cf. A002144, A033203, A007519, A004767, A016813, A005843.
%K A214588 nonn
%O A214588 1,1
%A A214588 _Brad Clardy_, Jul 22 2012