A214588 - OEIS

%I #21 Jul 26 2024 06:40:08

%S 2,3,5,7,17,19,23,37,53,67,71,83,97,101,103,113,131,149,151,163,167,

%T 179,181,193,197,199,211,227,229,241,257,263,277,293,307,311,337,353,

%U 359,373,389,401,419,421,433,439,449,467,487,499,503,547,563,577,593,599

%N Primes p such that p mod 16 < 8.

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

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

%C 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 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 Indranil Ghosh, <a href="/A214588/b214588.txt">Table of n, a(n) for n = 1..10000</a>

%e 103 is in the sequence because 103 mod 16 is 7 which is less than 8. - _Indranil Ghosh_, Jan 18 2017

%t Select[Prime[Range[200]],Mod[#,16]<8&] (* _Harvey P. Dale_, Jan 11 2018 *)

%o (Magma)

%o XOR := func<a, b | Seqint([ (adigs[i] + bdigs[i]) mod 2 : i in [1..n]], 2)

%o                        where adigs := Intseq(a, 2, n)

%o                        where bdigs := Intseq(b, 2, n)

%o                        where n := 1 + Ilog2(Max([a, b, 1]))>;

%o for n in [2 .. 1000] do

%o    if IsPrime(n)  then  pn:=n;

%o       if (XOR(pn,8) eq pn+8) then pn; end if;

%o    end if;

%o end for;

%o (PARI) is_A214588(p)={ !bittest(p,3) & isprime(p) } \\ _M. F. Hasler_, Jul 24 2012

%o (PARI) forprime(p=1,699, bittest(p,3) || print1(p",")) \\ _M. F. Hasler_, Jul 24 2012

%o (Python)

%o from sympy import isprime

%o i=1

%o j=1

%o while j<=10000:

%o     if  isprime(i)==True and (i%16)<8:

%o         print str(j)+" "+str(i)

%o         j+=1

%o     i+=1 # _Indranil Ghosh_, Jan 18 2017

%Y Cf. A002144, A033203, A007519, A004767, A016813, A005843.

%K nonn

%O 1,1

%A _Brad Clardy_, Jul 22 2012