cp's OEIS Frontend

For all i, j: A322809(i) = A322809(j) <=> a(i+1) = a(j+1).

For all i, j: a(i) = a(j) => b(i) = b(j), where b can be, but is not limited to, any of the following sequences: A029834, A049084, A062590, A063377, A064891, A078442 (A049076), A175663, A175682, A269668, A292936, A323162, many of which are related to counting primes in certain kinds of chains or iterations.

Why does this work? Consider the function f given in the definition: based on its properties, we can deduce from the value of f(n) the following information about n:

(A) If f(n) = -2, then n is 2, the only even prime,

(B) If f(n) = -3, then n is 3, the first odd prime,

(C) If f(n) is zero, then n is an even composite preceded by a prime, but we don't know which even composite exactly,

(D) If f(n) > 0 and f(1+2*f(n)) = f(2+2*f(n)), then n is either (D1) an odd composite number, or (D2) an even composite number preceded by an odd composite number, and the said composite number in both cases is 1 + 2*f(n),

(E) If f(n) > 0 and f(1+2*f(n)) <> f(2+2*f(n)), then n is an odd prime > 3, specifically, 1 + 2*f(n).

As this sequence is a restricted growth sequence transform of the said function f, we have a(i) = a(j) <=> f(i) = f(j) for all i, j, thus, even without knowing the value of n, but just a(n), we can find the value of f(n) by searching for the minimal k such that a(k) = a(n), then compute f(k) with that k. Furthermore, any function g defined as g(n) = h(f(n)) [where h is any function], clearly satisfies

a(i) = a(j) => g(i) = g(j), for all i, j. [*]

For instances of such functions g, we can consider many sequences like those sequences b(n) listed above, that have g(n) = 0 for all composite numbers, and g(p) > 0 for all primes p. This is usually the pattern, but there are exceptions, like A323162, which is the characteristic function of A005381, composites n such that n-1 is also composite. These are precisely the numbers that occur twice in this sequence, while all other numbers (including primes), occur just once, that is, reside in their own singular equivalence classes. Thus, it is not guaranteed that all sequences g matching to this sequence (i.e. those satisfying the implication *), even if not false positives in strict sense, would necessarily have some consistent relation to primes, instead, they might contain any random values at the positions given by A093515. However, in the current OEIS, such sequences are exceedingly rare.

Most prime numbers are absent; from first 1000 primes only 138 are terms: 3, 11, 17, 41, 67, 83, 109, 127, 157, 191, 211, 241, 277, 283, 353, ..., all with prime indices. Also, all nonprimes are terms. - Zak Seidov, Sep 12 2015

k is a term if and only if A078442(k) is even. - Pontus von Brömssen, Jul 25 2025

This sequence is the result of applying the N-sieve to generate the prime number subsequence A333242 where 1 indicates a prime number chosen to be included in sequence A333242 and 0 indicates the prime numbers and composites not in A333242.

This sequence can also be generated by the N-sieve.