A378619 -id:A378619 - cp's OEIS frontend

A112925 Largest squarefree integer < the n-th prime.

1, 2, 3, 6, 10, 11, 15, 17, 22, 26, 30, 35, 39, 42, 46, 51, 58, 59, 66, 70, 71, 78, 82, 87, 95, 97, 102, 106, 107, 111, 123, 130, 134, 138, 146, 149, 155, 161, 166, 170, 178, 179, 190, 191, 195, 197, 210, 222, 226, 227, 231, 238, 239, 249, 255, 262, 267, 269, 274, 278

Offset: 1

Leroy Quet, Oct 06 2005

nonn

			6 is the largest squarefree less than the 4th prime, 7. So a(4) = 6.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

For prime powers instead of squarefree numbers we have A065514, opposite A345531.
Restriction of A070321 (differences A378085) to the primes; see A378619.
The opposite is A112926, differences A378037.
Subtracting each term from prime(n) gives A240473, opposite A240474.
For nonsquarefree numbers we have A378033, differences A378036, see A378034, A378032.
For perfect powers we have A378035.
First differences are A378038.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259.
A013928 counts squarefree numbers up to n - 1.
A013929 lists the nonsquarefree numbers, differences A078147.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
A112929 counts squarefree numbers up to prime(n).
Cf. A061400, A112928, A112930.
Cf. A007674, A007947, A045882, A053797, A053806, A067535, A072284, A073247, A081217, A175216, A280892, A377783.

with(numtheory): a:=proc(n) local p,B,j: p:=ithprime(n): B:={}: for j from 1 to p-1 do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: B[nops(B)] end: seq(a(m),m=1..75); # Emeric Deutsch, Oct 14 2005

With[{k = 120}, Table[SelectFirst[Range[Prime@ n - 1, Prime@ n - Min[Prime@ n - 1, k], -1], SquareFreeQ], {n, 60}]] (* Michael De Vlieger, Aug 16 2017 *)

a(n,p=prime(n))=while(!issquarefree(p--),); p \\ Charles R Greathouse IV, Aug 16 2017

a(n) = prime(n) - A240473(n). - Gus Wiseman, Jan 10 2025

More terms from Emeric Deutsch, Oct 14 2005

A070321 Greatest squarefree number <= n.

Original entry on oeis.org

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

Benoit Cloitre, May 11 2002

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).

			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)

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]

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.

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

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 *)

a(n) = while (! issquarefree(n), n--); n; \\ Michel Marcus, Mar 18 2017

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

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

a(n) = A005117(A013928(n+1)). - Ridouane Oudra, Jul 26 2025

New description from Reinhard Zumkeller, Oct 03 2002

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A112925 Largest squarefree integer < the n-th prime.

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