login
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).
3
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
Sequence in context: A252448 A021992 A337123 * A261035 A080030 A337589
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, Sep 15 2017
STATUS
approved