cp's OEIS Frontend

a(n) purportedly gives the least k with A023193(k) = n; that is, this sequence should be the "least inverse" of A023193.

My web page extends the sequence to rho(305)=2047 and also gives a super-dense occurrence at rho(592)=4333 when pi(4333)=591 - the first known occurrence. - Thomas J Engelsma (tom(AT)opertech.com), Feb 16 2004

Tomás Oliveira e Silva (see link) has a table extending to n = 1000.

The minimal y such that there are n elements of {1, ..., y} with fewer than p distinct elements mod p for all prime p. - Charles R Greathouse IV, Jun 13 2013

The (wrong) old name was: Largest number of pairwise coprime numbers that can occur in an interval of length n. - Wolfdieter Lang, Oct 10 2017

Conjecturally, a(n) is the largest number of primes that occurs on an infinite number of intervals of n consecutive integers. The conjecture is apparently due to Dickson; Hardy & Littlewood's Conjecture B concerns only pairs (p, p + 2n).

According to the link at www.opertech.com, a(3159) >= 447 > 446 = pi(3159). The k-tuples conjecture then implies that, for an infinitude of n, the interval [n+1, n+3159] includes 447 primes. For these n, pi(n+3159) >= pi(n)+447 > pi(n)+446 = pi(n)+pi(3159), contradicting the conjecture that pi(x+y) <= pi(x)+pi(y). - David W. Wilson, May 23 2005

From Wolfdieter Lang, Oct 10 2017: (Start)

The following admissibility definition is adapted from the Hensley and Richards [H-R] or Richards [R] links. A k-tuple B_k = [b_1, b_2, ..., b_k] of integers with 0 <= b_1 < b_2 < ... < b_k is admissible if, for each prime p, there exists at least one congruence class modulo p which contains none of the B_k members. Because complete residue systems modulo p are equivalent under translation one can consider the length n interval [0, 1, ..., n-1] and admissible k-tuples starting with 0. The prime p = 2 allows then only even tuple numbers from I_n = [0, 2, ..., floor((n-1)/2)]. Only primes p <= k have to be tested.

a(n) is then the maximal k for which there is at least one such admissible B_k tuple from the interval I_n. This function a(n) is called rho^* in (H-R) and (R). It has been given as rhobar in the Schinzel - Sierpiński link, Théorème 1, p. 201.

Note that there are also admissible k-tuples from members of [0, 1, ..., n-1] which do not start with 0. Such tuples are translations of the ones starting with 0. E.g., [1, 3] is an admissible 2-tuple for any [0, 1,..., n-1] interval with n >= 3, but it is a translation of the considered [0, 2] tuple.

For the multiplicities of k see A047947(k), for k >= 1.

For the smallest k such that a(k) = n see A020497(n), for k >= 1.

For the number of all admissible k-tuples from the interval I_n starting with 0 see the array A292224(n, k), with k = 1..a(n), which has been given in the Engelsma link, Table 2, p. 27.

One of the Hardy-Littlewood conjectures (the prime tuple conjecture, see also conjecture (B) given by [H-R] and [R], and Ribenboim, hypothesis (D_1), p. 373, from the Dickson conjecture) is that there are infinitely many primes with gaps defined by any admissible B_k tuple, that is, all p, p + b_2, ..., p + b_k are prime for infinitely many primes p, for k >= 2. For k = 1 this is well known.

(End)

For a proper definition see the cross-references.