A293927 - OEIS

%I #25 Oct 23 2017 01:14:29

%S 17,28,30,33,36,43,45,47,51,52,56,58,65,66,72,74,76,80,84,90,94,107,

%T 111,119,126,129,130,133,137,143,145,155,156,166,169,174,179,185,192,

%U 200,202,204,208,213,214,216,219,228,238,246,248,249,250,254,258,262

%N Numbers n such that prime(k) XOR prime(k+1) XOR ... XOR prime(n) = 0 for some k < n (where XOR denotes the binary XOR operator, and prime(n) = A000040(n)).

%C Equivalently, numbers n such that A126084(n) = A126084(m) for some m < n.

%C See A293983(n) for the least k such that prime(k) XOR prime(k+1) XOR ... XOR prime(a(n)) = 0.

%H Robert Israel, <a href="/A293927/b293927.txt">Table of n, a(n) for n = 1..10000</a>

%e prime(33) XOR prime(34) XOR prime(35) XOR prime(36) = 137 XOR 139 XOR 149 XOR 151 = 0, hence 36 appears in the sequence.

%p N:= 1000: # to get all terms <= N

%p R[0]:= 0: T:= 2: p:= 2;

%p Res:= NULL:

%p for n from 2 to N do

%p   p:= nextprime(p);

%p   T:= Bits:-Xor(T,p);

%p   if assigned(R[T]) then Res:= Res, n

%p   else R[T]:= n

%p   fi

%p od:

%p Res; # _Robert Israel_, Oct 22 2017

%o (PARI) s = 0; seen = 2^0; for (i = 1, 262, s = bitxor(s, prime(i)); if (bittest(seen, s), print1 (i ", "), seen += 2^s))

%Y Cf. A000040, A126084, A293983.

%K nonn,base

%O 1,1

%A _Rémy Sigrist_, Oct 21 2017