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2, 3, 7, 13, 5, 17, 31, 11, 29, 19, 41, 67, 23, 47, 79, 37, 53, 83, 43, 71, 107, 59, 97, 131, 61, 113, 163, 73, 127, 173, 89, 149, 211, 101, 157, 223, 103, 167, 109, 181, 257, 139, 227, 307, 137, 229, 151, 233, 331, 179, 281, 349, 191, 277, 373, 193, 293, 199, 311, 197, 271, 379, 239, 367, 241, 347, 463, 251, 383, 487, 263, 397, 521, 269, 419, 283, 421, 563, 313, 443, 587

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proposed

After a(1) = 2, we cannot have a(2) = 1 as 1 is not a prime number; a(2) = 3 is ok OK as the absolute difference |2-3| = 1 is a nonprime; the next term a(3) cannot be 5 as the absolute difference |3-5| = 2 is a prime (and we don't want primes in the absolute differences); a(3) = 7 is ok OK as the absolute difference |3-7| = 4 is a nonprime not yet present in the absolute differences; the next term a(4) cannot be 5 as the absolute difference |7-5| = 2 is a prime; the next term a(4) cannot be 11 as the absolute difference |7-11| = 4 is already in the absolute differences, a(4) = 13 is ok OK as the absolute difference |7-13| = 6 is a nonprime not yet present in the absolute differences; the next term a(5) is now 5 as |13-5| = 8 is a nonprime not yet present in the absolute differences; the next term a(6) cannot be 11, the smallest available prime, as the absolute difference |5-11| = 6 is a nonprime already present in the absolute differences; a(6) = 17 is ok OK as |5-17| = 12 is a nonprime not yet present in the absolute differences; the next term a(7) cannot be 11, 19, 23 or 29 for one of the above reasons, but a(7) = 31 is ok OK as |17-31| = 14 is a nonprime not yet present in the absolute differences; etc.

The sequence starts with a(1) = 2 and was is always extended with the smallest integer not yet present that does not lead to a contradiction.

The equivalent sequence where nonprimes and primes exchange their roles is A277997.

A277997.

After a(1) = 2, we cannot have a(2) = 1 as 1 is not a prime number; a(2) = 3 is ok as the absolute difference |2-3| = 1 is a nonprime; the next term a(3) cannot be 5 as the absolute difference |3-5| = 2 is a prime (and we don’'t want primes in the absolute differences); a(3) = 7 is ok as the absolute difference |3-7| = 4 is a nonprime not yet present in the absolute differences; the next term a(4) cannot be 5 as the absolute difference |7-5| = 2 is a prime; the next term a(4) cannot be 11 as the absolute difference |7-11| = 4 is already in the absolute differences, a(4) = 13 is ok as the absolute difference |7-13| = 6 is a nonprime not yet present in the absolute differences; the next term a(5) is now 5 as |13-5| = 8 is a nonprime not yet present in the absolute differences; the next term a(6) cannot be 11, the smallest available prime, as the absolute difference |5-11| = 6 is a nonprime already present in the absolute differences; a(6) = 17 is ok as |5-17| = 12 is a nonprime not yet present in the absolute differences; the next term a(7) cannot be 11, 19, 23 or 29 for one of the above reasons, but a(7) = 31 is ok as |17-31| = 14 is a nonprime not yet present in the absolute differences; etc.

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The equivalent sequence where nonprimes and primes exchange their roles is

A277997.

The equivalent sequence where nonprimes and primes exchange their roles is

Cf. A277997.

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proposed

The equivalent sequence where nonprimes and primesexchange primes exchange their roles is