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The Two-Up sequence: a(n) is the least positive number not already used that is coprime to the previous floor(n/2) terms.
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1, 2, 3, 5, 4, 7, 9, 11, 13, 17, 8, 19, 23, 25, 21, 29, 31, 37, 41, 43, 47, 53, 16, 59, 61, 67, 71, 73, 55, 79, 27, 49, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 26, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 85, 121, 223, 227, 57, 229
a(n) = binary expansion of A354169(n).
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0, 1, 10, 100, 1000, 11, 10000, 100000, 1000000, 1100, 10000000, 100000000, 1000000000, 10001, 10000000000, 100010, 100000000000, 1000000000000, 10000000000000, 1000100, 100000000000000, 10001000, 1000000000000000, 10000000000000000, 100000000000000000, 1100000000, 1000000000000000000, 10000000000000000000, 100000000000000000000
Let A354790(n) = Product_{i >= 1} prime(i)^e(i); then a(n) is the concatenation, in reverse order, of e_1, e_2, ..., ending at the exponent of the largest prime factor of A354790(n); a(1)=0 by convention.
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0, 1, 10, 100, 1000, 10000, 11, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 1100, 10001, 100000000000, 100010, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 10000000000000000, 100000000000000000, 1000000000000000000, 10000000000000000000
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