OFFSET
1,4
COMMENTS
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
LINKS
Brian Kehrig, Table of n, a(n) for n = 1..1000
Brian Kehrig, Python code for sequence
EXAMPLE
Between two multiples of 15 (n and n+15), only n+1, n+2, n+4, n+7, n+8, n+11, n+13, and n+14 could possibly be prime based on divisibility by 3 and 5. However, 4 of these are even and 4 are odd, so at most 4 of them can be prime. Thus, a(15)=4.
PROG
(Python) # see Links section
CROSSREFS
KEYWORD
nonn
AUTHOR
Brian Kehrig, Aug 24 2023
STATUS
approved