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A290411 Maximum length of a run of consecutive positive integers that are either squarefree or divisible by the square of at least one of the first n primes. 0
3, 17, 74, 195, 675, 1182, 2168, 3609 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The length of a run of consecutive squarefree positive integers is limited to 3 because no multiple of 4 is squarefree; thus, a(0) <= 3.

But suppose we consider runs of consecutive positive integers that are either squarefree or divisible by 4. No such set of 18 numbers can exist, since it would include two multiples of 3^2 = 9, at most one of which would be divisible by 4, so a(1) <= 17.

No set of 75 consecutive positive integers can have every member squarefree or divisible by 4 or 9, since any such set would include three multiples of 5^2 = 25, at most two of which would be divisible by 4 or 9, so a(2) <= 74.

By a similar argument, a(3) <= 4*7^2 - 1 = 195.

However, it does not follow in general that a(n) <= (n+1)*prime(n+1)^2 - 1; e.g., a(4) = 675 > 605 = 5*11^2, and the five multiples of 121 in the interval [2070582452238, 2070582452912], i.e., 121*{17112251672..17112251676}, are also divisible by 4, 9, 49, 25, and 4, respectively.

For n = 0..4, the smallest positive k that begins a run of a(n) consecutive integers of which each is squarefree or divisible by the square of at least one of the first n primes is 1, 28, 28976, 64779, and 2070582452238, respectively.

LINKS

Table of n, a(n) for n=0..7.

EXAMPLE

1, 2, and 3 are all squarefree, and no four consecutive positive integers can be squarefree, so a(0) = 3.

Each of the 17 integers in {28, ..., 44} is either squarefree or divisible by 2^2 = 4, and no interval of more than 17 consecutive positive integers has this property, so a(1) = 17.

Each of the 74 consecutive positive integers in {28976, ..., 29049} is squarefree or divisible by 4 or 9, and no larger set has this property, so a(2) = 74.

CROSSREFS

Sequence in context: A179596 A339647 A191587 * A142868 A171875 A356390

Adjacent sequences: A290408 A290409 A290410 * A290412 A290413 A290414

KEYWORD

nonn,more

AUTHOR

Jon E. Schoenfield, Jul 30 2017

EXTENSIONS

a(5)-a(7) from Jinyuan Wang, Jun 24 2022

STATUS

approved

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Last modified December 7 17:25 EST 2022. Contains 358668 sequences. (Running on oeis4.)