A070321 - OEIS (original) (raw)

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

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).

EXAMPLE

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

local a;

for a from n by -1 do

if issqrfree(a) then

return a;

end if;

end do:

end proc:

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

The distinct terms are A005117 (the squarefree numbers).

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.