A070321 Greatest squarefree number <= n.

1, 2, 3, 3, 5, 6, 7, 7, 7, 10, 11, 11, 13, 14, 15, 15, 17, 17, 19, 19, 21, 22, 23, 23, 23, 26, 26, 26, 29, 30, 31, 31, 33, 34, 35, 35, 37, 38, 39, 39, 41, 42, 43, 43, 43, 46, 47, 47, 47, 47, 51, 51, 53, 53, 55, 55, 57, 58, 59, 59, 61, 62, 62, 62, 65, 66, 67, 67, 69, 70, 71, 71

Offset

1,2

Comments

a(n) = Max( core(k) : k=1,2,3,...,n ) where core(x) is the squarefree part of x (the smallest integer such that x*core(x) is a square).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Mayank Pandey, Squarefree numbers in short intervals, arXiv preprint (2024). arXiv:2401.13981 [math.NT]

Formula

a(n) = n - o(n^(1/5)) by a result of Pandey. - Charles R Greathouse IV, Dec 04 2024

Example

From Gus Wiseman, Dec 10 2024: (Start)
The squarefree numbers <= n are the following columns, with maxima a(n):
  1  2  3  3  5  6  7  7  7  10  11  11  13  14  15  15
     1  2  2  3  5  6  6  6  7   10  10  11  13  14  14
        1  1  2  3  5  5  5  6   7   7   10  11  13  13
              1  2  3  3  3  5   6   6   7   10  11  11
                 1  2  2  2  3   5   5   6   7   10  10
                    1  1  1  2   3   3   5   6   7   7
                             1   2   2   3   5   6   6
                                 1   1   2   3   5   5
                                         1   2   3   3
                                             1   2   2
                                                 1   1
(End)

Maple

A070321 := proc(n)
    local a;
    for a from n by -1 do
        if issqrfree(a) then
            return a;
        end if;
    end do:
end proc:
seq(A070321(n), n=1..100) ; # R. J. Mathar, May 25 2023

Mathematica

a[n_] :=For[ k = n, True, k--, If[ SquareFreeQ[k], Return[k]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 27 2013 *)
gsfn[n_]:=Module[{k=n}, While[!SquareFreeQ[k], k--]; k]; Array[gsfn, 80] (* Harvey P. Dale, Mar 27 2013 *)

Prog

(PARI) a(n) = while (! issquarefree(n), n--); n; \\ Michel Marcus, Mar 18 2017
(Python)
from itertools import count
from sympy import factorint
def A070321(n): return next(m for m in count(n, -1) if max(factorint(m).values(), default=0)<=1) # Chai Wah Wu, Dec 04 2024

Crossrefs

Cf. A007947, A076260.

Cf. A081217, A081218, A081210.

The distinct terms are A005117 (the squarefree numbers).

The opposite version is A067535, differences A378087.

The run-lengths are A076259.

Restriction to the primes is A112925; see A378038, A112926, A378037.

For nonsquarefree we have A378033; see A120327, A378036, A378032, A377783.

First differences are A378085.

Subtracting each term from n gives A378619.

A013929 lists the nonsquarefree numbers, differences A078147.

A061398 counts squarefree numbers between primes, zeros A068360.

A061399 counts nonsquarefree numbers between primes, zeros A068361.

Cf. A007674, A013928, A053797, A053806, A072284, A073247, A112929, A240473.

Sequence in context: A081213 A081210 A285719 * A378363 A239904 A334819

Adjacent sequences: A070318 A070319 A070320 * A070322 A070323 A070324

Keyword

easy,nonn

Author

Benoit Cloitre, May 11 2002

Extensions

New description from Reinhard Zumkeller, Oct 03 2002

Status

approved