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a(n) is the binary XOR of all n-bit prime numbers.
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%I #19 May 09 2021 07:56:20

%S 0,1,2,6,23,59,99,203,469,807,1615,3349,2266,4576,14042,25002,89193,

%T 131215,135904,814531,885682,60842,3969154,3370892,6742296,14350136,

%U 42766902,97565102,444197631,515121776,2085329975,2091732354,7999937231,14794305847

%N a(n) is the binary XOR of all n-bit prime numbers.

%C XOR is the binary exclusive-or operator.

%C a(1)=0 for compatibility with similar sequences, and because 0 and 1 are not primes.

%C Note the sequence s(n)-a(n), where s(n)=A298816(n) is the binary XOR of all n-bit squares, begins: 1, -1, 2, 3, -14, -38, -87, -175, -20, -230, -1258, -2352, 3819, 9957, -1525, -9925, 31932, 21654, 264124, 226521, 405022, 2495526, 944510, 8579700, 15679080, 49342536, -35092149, -19209773, -131473914. The distribution of negative and positive terms does not look random: runs of negative terms are followed by runs of positive terms.

%H Lars Blomberg, <a href="/A298817/b298817.txt">Table of n, a(n) for n = 1..41</a>

%e There are two 4-bit primes, namely 11 and 13. a(4) = (11 XOR 13) = 6.

%o (Python)

%o from sympy import nextprime

%o n = x = L = 2

%o print('0', end=',')

%o while L < 27:

%o nextn = nextprime(n)

%o if (nextn ^ n) > n: # if lengths of binary representations are different

%o print(str(x), end=',')

%o x = 0

%o prevL = L

%o L = len(bin(nextn))-2

%o for j in range(prevL, L-1): print('0', end=',')

%o n = nextn

%o x ^= n

%o (PARI) a(n) = {my(x = 0); for (k=2^(n-1), 2^n-1, if (isprime(k), x = bitxor(x, k));); x;} \\ _Michel Marcus_, Jan 27 2018

%Y Cf. A000040, A007088, A070939, A035100, A298816.

%Y Cf. also A014234, A104080.

%K nonn,base

%O 1,3

%A _Alex Ratushnyak_, Jan 26 2018

%E a(30)-a(34) from _Lars Blomberg_, Nov 10 2018