cp's OEIS Frontend

The old name was: "Schinzel's rhobar(n), number of distinct lengths of a block of consecutive integers on which a maximum of n primes occurs infinitely often (under the k-tuple conjecture)." [Note that "rhobar" is A023193.]

3 patterns for 8-tuplets: 11010011001011, 11011010011001 and v.v.

See A022011, A022012 and A022013 for the three different possible patterns. The sequence is conjectured to be infinite, although it is not even proved that there are infinitely many twin primes (p1, p2 = p1+2). - M. F. Hasler, May 02 2015

With increasing n the radius of log(n) slowly increases, while frequency of prime-powers decreases. Thus hard to estimate upper bound of terms in this sequence.

Heuristically a(n) = 0 about 1/e^2 = 13.53...% of the time. The first few instances are 1, 300, 324, 895, 896, 897, 898, 899, 1077, .... - Charles R Greathouse IV, Apr 30 2015

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

For a proper definition see the cross-references.

Alternatively: a(n) = the maximum number of elements of an admissible k-tuple strictly contained in (0,n) such that all elements are relatively prime to n. Recall that an admissible tuple is defined as a tuple of integers with the property that all primes p have at least one residue class that has no intersection with the tuple.

For n > 1, we have a(n) <= A023193(n-1), with equality if (but not only if) n is prime or a power of 2. The smallest n for which it is not an equality is n=14.

Conjecture 1: Every nonnegative integer appears in this sequence.

Conjecture 2: For all n, there is an infinitude of k's such that there are a(n) primes between n*k and n*(k+1).

Conjecture 2 resembles the k-tuples conjecture a.k.a. the first Hardy-Littlewood conjecture, although it is not the same.

A notable value is a(35) = 8. Compare with A000010(210) = 48. This says that between any two consecutive multiples of 210 the 48 values that are not divisible by 2, 3, 5 or 7 are equally distributed between 6 equal divisions of 210; that is, 8 are in the interval [0, 34], 8 in the interval [35, 69], etc. - Peter Munn, Feb 16 2024

The row lengths are given by A023193 (the rhobar function of Schinzel and Sierpiński called rho* by Hensley and Richards).

This irregular triangle has already been considered by Engelsma, see Table 2, for n=1..56, p. 27.

A k-tuple of integers B_k = [b_1, ..., b_k] with 0 = b_1 < b_2 < ... < b_k <= n-1 is called admissible if for each prime p there exists at least one congruence class modulo p which contains none of the B_k elements. (This corresponds to the alternative definition of Hensley and Richards, p. 378 (*) or Richards, p. 423, 1.5 Definition and (*).) Note that the definition of "admissibility" is translation invariant (see the Note by Richards, p. 424, which is obvious from the translation equivalence of complete residue systems modulo p). Therefore the interval I_n = [0, n-1] of length n has been chosen. The b_1 = 0 choice is conventional. Without this choice other admissible k-tuples are obtained by translation as long as b_k + a < n-1. E.g., for n = 8 and k = 3 the tuple [1, 5, 7] is admissible and a translation of the considered tuple [0, 4, 6].

Only primes p <= k have to be tested to decide on the admissibility of a B_k tuple because for larger k there is always some residue class which contains none of the k members of B_k.

Because p = 2 already forbids even and odd numbers to appear in B_k for k >= 2, one can for the admissibility test eliminate all odd numbers in the chosen I_n. Therefore, only Ieven_n:= [0, 2, ..., 2*floor((n-1)/2)] =: 2*[0, 1, ..., floor((n-1)/2)] need be considered. B_1 = [0] is admissible for all n >= 1.

Because only the interval Ieven_n is of relevance, there will occur repetitions for admissible tuples for n if n = 2*k+1 and n = 2*k+2.

With the set B_k(p) = B_k (mod p) := {0, b_1 (mod p), ..., b_k (mod p)} the criterion for admissibility can be written as p - #(B_k(p)) > 0, for all primes 3 <= p <= k (because there are p congruence classes defined by smallest nonnegative complete residue system [0, 1, ..., p-1]).

Admissible tuples (starting with 0) with least b_k - b_1 = b_k value give rise to prime k-constellations of diameter b_k. E.g., for k = 2 the admissible tuple [0, 4] does not lead to a prime 2-constellation for n >= 5; [0, 6] is out for n >= 7, ... . But there are two prime 3-constellations given by [0, 2, 6] and [0, 4, 6] for n >= 7.

Row sums are in A292225, that is, total number of admissible tuples starting with 0 from the interval I_n = [0, n-1].

This sequence is given in column 2 of Table 2, p. 27, of the Engelsma link.

See A292224 for the reason for the repetitions for n = 2*k+1 and n = 2*(k+1) for k >= 0, the definition of "admissible", references, and examples of these admissible k-tuples for n = 1..10 (with k = 1, 2, ..., A023193(n)).