A070321 - OEIS

%I #43 Jul 26 2025 08:06:56

%S 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,

%T 26,26,29,30,31,31,33,34,35,35,37,38,39,39,41,42,43,43,43,46,47,47,47,

%U 47,51,51,53,53,55,55,57,58,59,59,61,62,62,62,65,66,67,67,69,70,71,71

%N Greatest squarefree number <= n.

%C 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).

%H Reinhard Zumkeller, <a href="/A070321/b070321.txt">Table of n, a(n) for n = 1..10000</a>

%H Mayank Pandey, <a href="https://arxiv.org/abs/2401.13981">Squarefree numbers in short intervals</a>, arXiv preprint (2024). arXiv:2401.13981 [math.NT]

%F a(n) = n - o(n^(1/5)) by a result of Pandey. - _Charles R Greathouse IV_, Dec 04 2024

%F a(n) = A005117(A013928(n+1)). - _Ridouane Oudra_, Jul 26 2025

%e From _Gus Wiseman_, Dec 10 2024: (Start)

%e The squarefree numbers <= n are the following columns, with maxima a(n):

%e   1  2  3  3  5  6  7  7  7  10  11  11  13  14  15  15

%e      1  2  2  3  5  6  6  6  7   10  10  11  13  14  14

%e         1  1  2  3  5  5  5  6   7   7   10  11  13  13

%e               1  2  3  3  3  5   6   6   7   10  11  11

%e                  1  2  2  2  3   5   5   6   7   10  10

%e                     1  1  1  2   3   3   5   6   7   7

%e                              1   2   2   3   5   6   6

%e                                  1   1   2   3   5   5

%e                                          1   2   3   3

%e                                              1   2   2

%e                                                  1   1

%e (End)

%p A070321 := proc(n)

%p     local a;

%p     for a from n by -1 do

%p         if issqrfree(a) then

%p             return a;

%p         end if;

%p     end do:

%p end proc:

%p seq(A070321(n),n=1..100) ; # _R. J. Mathar_, May 25 2023

%t a[n_] :=For[ k = n, True, k--, If[ SquareFreeQ[k], Return[k]]]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Mar 27 2013 *)

%t gsfn[n_]:=Module[{k=n},While[!SquareFreeQ[k],k--];k]; Array[gsfn,80] (* _Harvey P. Dale_, Mar 27 2013 *)

%o (PARI) a(n) = while (! issquarefree(n), n--); n; \\ _Michel Marcus_, Mar 18 2017

%o (Python)

%o from itertools import count

%o from sympy import factorint

%o 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

%Y Cf. A007947, A076260.

%Y Cf. A081217, A081218, A081210.

%Y The distinct terms are A005117 (the squarefree numbers).

%Y The opposite version is A067535, differences A378087.

%Y The run-lengths are A076259.

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

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

%Y First differences are A378085.

%Y Subtracting each term from n gives A378619.

%Y A013929 lists the nonsquarefree numbers, differences A078147.

%Y A061398 counts squarefree numbers between primes, zeros A068360.

%Y A061399 counts nonsquarefree numbers between primes, zeros A068361.

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

%K easy,nonn

%O 1,2

%A _Benoit Cloitre_, May 11 2002

%E New description from _Reinhard Zumkeller_, Oct 03 2002