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%I #32 May 08 2021 08:29:35
%S 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,
%T 49,50,51,52,53,54,59,60,62,63,64,65,67,68,69,70,71,72,73,79,83,89,97,
%U 101,103,107,109,113,127,131,132,133,134,135,137,139,140,149
%N 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.
%C 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.
%C 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.
%C 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.
%e a(0) cannot be 0, since then 0 should be prime, which it is not.
%e a(0) = 1 is valid hence a(1) must be the next prime, which is a(1) = 2.
%e Then a(2) should be the next prime, hence a(2) = 3.
%e a(3) should be prime, hence a(3) = 5.
%e Since 4 is not in the sequence so far, a(4) must be the next nonprime, which means a(4) = 6.
%o (Python)
%o # 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.
%o def a_list(length):
%o """Returns the list [a(0), ..., a(length-1)]."""
%o num = 1
%o b = [1]
%o for i in range(1, length):
%o num += 1
%o if i in b:
%o while not is_prime(num):
%o num += 1
%o b.append(num)
%o else:
%o while is_prime(num):
%o num += 1
%o b.append(num)
%o return b
%o print(a_list(63))
%Y The same definition with A079254 except indices start from 0 instead of 1.
%Y Cf. A079000, A079313.
%K nonn
%O 0,2
%A _Adnan Baysal_, Apr 13 2020
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