A298817 a(n) is the binary XOR of all n-bit prime numbers.

0, 1, 2, 6, 23, 59, 99, 203, 469, 807, 1615, 3349, 2266, 4576, 14042, 25002, 89193, 131215, 135904, 814531, 885682, 60842, 3969154, 3370892, 6742296, 14350136, 42766902, 97565102, 444197631, 515121776, 2085329975, 2091732354, 7999937231, 14794305847

Offset: 1

Author

Alex Ratushnyak, Jan 26 2018

Keywords

• nonn
• base

Comments

XOR is the binary exclusive-or operator.

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

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.

Examples

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

Links

• Lars Blomberg, Table of n, a(n) for n = 1..41

Crossrefs

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

Cf. also A014234, A104080.

Programs

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

Python

from sympy import nextprime
n = x = L = 2
print('0', end=',')
while L < 27:
    nextn = nextprime(n)
    if (nextn ^ n) > n:  # if lengths of binary representations are different
        print(str(x), end=',')
        x = 0
        prevL = L
        L = len(bin(nextn))-2
        for j in range(prevL, L-1):  print('0', end=',')
    n = nextn
    x ^= n

Extensions

a(30)-a(34) from Lars Blomberg, Nov 10 2018