A375828 -id:A375828 - cp's OEIS frontend

A375829 a(1) = 1; for n > 1, a(n) is the smallest unused positive number such that (a(n-1) AND a(n)) = 0 if a(n-1) is prime, otherwise (a(n-1) AND a(n)) = a(n-1), where AND is the binary AND operation.

1, 3, 4, 5, 2, 8, 9, 11, 16, 17, 6, 7, 24, 25, 27, 31, 32, 33, 35, 39, 47, 64, 65, 67, 12, 13, 18, 19, 36, 37, 10, 14, 15, 63, 127, 128, 129, 131, 20, 21, 23, 40, 41, 22, 30, 62, 126, 254, 255, 511, 1023, 2047, 4095, 8191, 8192, 8193, 8195, 8199, 8207, 8223, 8255, 8319, 8447, 256, 257, 26, 58, 59, 68, 69, 71, 48, 49, 51, 55, 119, 247, 503, 512, 513, 515, 519

Offset: 1

Scott R. Shannon, Aug 30 2024

nonn

The long term behavior of the terms is dominated by the appearance of the Mersenne primes. This is due to composite numbers appearing which are 1 less than a power of 2, i.e., their binary representation consists of all 1's. Therefore the next term must be 1 less than the next power of 2 to satisfy (a(n-1) AND a(n)) = a(n-1). This pattern repeats until such a number is prime, i.e., a Mersenne prime, at which point the next term will be 1 more than this prime, after which the terms can eventually return to smaller values.

Due to the above behavior it is unknown if all numbers eventually appear, but assuming the number of Mersenne primes is infinite, it is likely all eventually will. In the first 100 terms the fixed points are 1, 71, 463, although more likely exist.

			a(7) = 9 as a(6) = 8 = 1000_2 is not prime, and ((9 = 1001_2) AND 1000_2) = 1000_2 = 8.
a(9) = 16 as a(8) = 11 = 1011_2 is prime, and ((16 = 10000_2) AND 1011_2) = 0.

Scott R. Shannon, Table of n, a(n) for n = 1..1100

Cf. A375828, A000668, A109812, A007088, A115510, A000040, A353989.

cp's OEIS Frontend

A375829 a(1) = 1; for n > 1, a(n) is the smallest unused positive number such that (a(n-1) AND a(n)) = 0 if a(n-1) is prime, otherwise (a(n-1) AND a(n)) = a(n-1), where AND is the binary AND operation.

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