A292388 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, SumXOR_{k=1..n} a(k) is prime (where SumXOR is the analog of summation under the binary XOR operation).

2, 1, 4, 5, 7, 6, 8, 9, 15, 10, 12, 14, 18, 16, 20, 17, 19, 22, 24, 26, 32, 30, 36, 28, 38, 34, 40, 42, 44, 43, 21, 50, 39, 29, 48, 45, 31, 52, 46, 58, 54, 62, 55, 41, 56, 60, 66, 68, 72, 74, 64, 78, 84, 76, 63, 57, 80, 86, 88, 94, 90, 92, 100, 70, 96, 98, 82

Offset: 1

Author

Rémy Sigrist, Sep 15 2017

Keywords

• nonn
• base

Comments

The partial XOR sums are given by A292389.

This sequence is similar to A054408: here we combine the first terms with the binary XOR operation, there with the classic sum operation.

There are no three consecutive odd terms.

If SumXOR_{k=1..n} a(k) = 2, then a(n+1) is odd.

Is this sequence a permutation of the natural numbers?

The only odd numbers that can appear have the form p XOR 2 for some prime p. Thus 3, 11, 13, 23, 25, 27, 33, 35, 37, 47, ... never appear. - Peter Munn, Jan 19 2023

Examples

			a(1) cannot equal 1 (1 is not prime).
a(1) = 2 is suitable.
a(2) = 1 is suitable.
a(3) cannot equal 1 (already used), 2 (already used) or 3 (2 XOR 1 XOR 3 = 0 is not prime).
a(3) = 4 is suitable.
a(4) cannot equal 1 (already used), 2 (already user), 3 (2 XOR 1 XOR 4 XOR 3 = 4 is not prime) or 4 (already used).
a(4) = 5 is suitable.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000 [third column removed by _Georg Fischer_, Mar 31 2023]

Crossrefs

Cf. A054408, A292389.

Programs

PARI

s=0; x=0; for (n=1, 67, for (v=1, oo, if (!bittest(s,v) && isprime(bitxor(x,v)), s+=2^v; x=bitxor(x,v); print1 (v ", "); break)))