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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).
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%I #26 Mar 31 2023 13:21:24

%S 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,

%T 34,40,42,44,43,21,50,39,29,48,45,31,52,46,58,54,62,55,41,56,60,66,68,

%U 72,74,64,78,84,76,63,57,80,86,88,94,90,92,100,70,96,98,82

%N 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).

%C The partial XOR sums are given by A292389.

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

%C There are no three consecutive odd terms.

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

%C Is this sequence a permutation of the natural numbers?

%C 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

%H Rémy Sigrist, <a href="/A292388/b292388.txt">Table of n, a(n) for n = 1..10000</a> [third column removed by _Georg Fischer_, Mar 31 2023]

%e a(1) cannot equal 1 (1 is not prime).

%e a(1) = 2 is suitable.

%e a(2) = 1 is suitable.

%e a(3) cannot equal 1 (already used), 2 (already used) or 3 (2 XOR 1 XOR 3 = 0 is not prime).

%e a(3) = 4 is suitable.

%e 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).

%e a(4) = 5 is suitable.

%o (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)))

%Y Cf. A054408, A292389.

%K nonn,base

%O 1,1

%A _Rémy Sigrist_, Sep 15 2017