A334067 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a term of the sequence if and only if a(n) is prime" where indices start from 0.

1, 2, 3, 5, 6, 7, 11, 13, 14, 15, 16, 17, 18, 19, 23, 29, 31, 37, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 59, 60, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 132, 133, 134, 135, 137, 139, 140, 149

Offset

0,2

Comments

a(n) is the minimal sequence for which the sequence generated by the indices of primes in this sequence is equal to itself, where indices start from 0.

So if f is a function on 0-indexed integer sequences with infinitely many primes where f returns the increasing sequence of indices of primes of the input sequence b(n), then a(n) is the lexicographically minimal fixed point of f.

a(n) has almost the same definition as A079254, except that a(n) starts indices from 0 instead of 1. But the resulting sequences do not seem to have any correlation.

Links

Table of n, a(n) for n=0..62.

Example

a(0) cannot be 0, since then 0 should be prime, which it is not.
a(0) = 1 is valid hence a(1) must be the next prime, which is a(1) = 2.
Then a(2) should be the next prime, hence a(2) = 3.
a(3) should be prime, hence a(3) = 5.
Since 4 is not in the sequence so far, a(4) must be the next nonprime, which means a(4) = 6.

Prog

(Python)
# is_prime(n) is a Python function which returns True if n is prime, and returns False otherwise. In the form stated below runs with SageMath.
def a_list(length):
    """Returns the list [a(0), ..., a(length-1)]."""
    num = 1
    b = [1]
    for i in range(1, length):
        num += 1
        if i in b:
            while not is_prime(num):
                num += 1
            b.append(num)
        else:
            while is_prime(num):
                num += 1
            b.append(num)
    return b
print(a_list(63))

Crossrefs

The same definition with A079254 except indices start from 0 instead of 1.

Cf. A079000, A079313.

Sequence in context: A288863 A121700 A080980 * A134669 A342392 A053328

Adjacent sequences: A334064 A334065 A334066 * A334068 A334069 A334070

Keyword

nonn

Author

Adnan Baysal, Apr 13 2020

Status

approved