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

From Phil Moore (moorep(AT)lanecc.edu), Dec 14 2009: (Start)

It is known that a(39278) = 0, since no such prime exists for the Sierpiński number 78557 (cf. A076336).

It has recently been discovered that 2131+2^4583176 and 41693+2^5146295 are probable primes, so a(1065) is probably 4583176 and a(20846) is probably 5146295.

At present, the only odd value less than 78557 for which no prime or strong probable prime of the form t+2^k is known is t = 40291, so a(20145) is completely unknown. In addition, for 25 values of t < 78557, only strong probable primes are known. (End)

The last case was resolved in 2011 when the probable prime 40291+2^9092392 was found as a part of a distributed project "Five or Bust". See links. - Jeppe Stig Nielsen, Mar 29 2019

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)