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

1,3

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.

Links

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

Example

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

Prog

(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
(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

Crossrefs

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

Cf. also A014234, A104080.

Sequence in context: A177252 A168270 A222761 * A245125 A290954 A220213

Adjacent sequences: A298814 A298815 A298816 * A298818 A298819 A298820

Keyword

nonn,base

Author

Alex Ratushnyak, Jan 26 2018

Extensions

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

Status

approved