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

1,1

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

Links

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

Example

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.

Prog

(PARI) s=0; x=0; for (n=1, 67, for (v=1, oo, if (!bit test(s, v) && is prime(bit xor(x, v)), s+=2^v; x=bit xor(x, v); print1 (v ", "); break)))

Crossrefs

Cf. A054408, A292389.

Sequence in context: A252448 A021992 A337123 * A261035 A080030 A337589

Adjacent sequences: A292385 A292386 A292387 * A292389 A292390 A292391

Keyword

nonn,base

Author

Rémy Sigrist, Sep 15 2017

Status

approved