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A333403
Lexicographically earliest sequence of positive integers such that for any m and n with m <= n, a(m) XOR ... XOR a(n) is neither null nor prime (where XOR denotes the bitwise XOR operator).
2
1, 8, 1, 48, 1, 8, 1, 68, 1, 8, 1, 48, 1, 8, 1, 1158, 1, 8, 1, 48, 1, 8, 1, 68, 1, 8, 1, 48, 1, 8, 1, 4752, 1, 8, 1, 48, 1, 8, 1, 68, 1, 8, 1, 48, 1, 8, 1, 1158, 1, 8, 1, 48, 1, 8, 1, 68, 1, 8, 1, 48, 1, 8, 1, 81926, 1, 8, 1, 48, 1, 8, 1, 68, 1, 8, 1, 48, 1, 8
OFFSET
1,2
COMMENTS
This sequence is a variant of A332941.
This sequence is infinite:
- suppose that the first n terms are known,
- let M = max_{k <= n} a(k) XOR ... XOR a(n),
- let k be such that M < 2^k,
- as there are prime gaps of any size,
we can choose an interval of the form [m*2^k..(m+1)*2^k] without prime numbers,
- hence a(n+1) <= m*2^k, QED.
LINKS
FORMULA
a(m) = a(n) iff A007814(n) = A007814(m).
a(n) = a(2^k-n) for any k >= 0 and n = 1..2^k-1.
EXAMPLE
The values of a(i) XOR ... XOR a(j) for i <= j <= 8 are:
i\j| 1 2 3 4 5 6 7 8
---+------------------------------
1| 1 9 8 56 57 49 48 116
2| . 8 9 57 56 48 49 117
3| . . 1 49 48 56 57 125
4| . . . 48 49 57 56 124
5| . . . . 1 9 8 76
6| . . . . . 8 9 77
7| . . . . . . 1 69
8| . . . . . . . 68
PROG
(PARI) See Links section.
CROSSREFS
Sequence in context: A286260 A226374 A050401 * A271060 A318576 A089276
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, Mar 22 2020
STATUS
approved