A005117 Squarefree numbers: numbers that are not divisible by a square greater than 1.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• core

Comments

1 together with the numbers that are products of distinct primes.

Also smallest sequence with the property that a(m)*a(k) is never a square for k != m. - Ulrich Schimke (ulrschimke(AT)aol.com), Dec 12 2001

Numbers k such that there is only one Abelian group with k elements, the cyclic group of order k (the numbers such that A000688(k) = 1). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 25 2001

Numbers k such that A007913(k) > phi(k). - Benoit Cloitre, Apr 10 2002

a(n) is the smallest m with exactly n squarefree numbers <= m. - Amarnath Murthy, May 21 2002

k is squarefree <=> k divides prime(k)# where prime(k)# = product of first k prime numbers. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 30 2004

Numbers k such that omega(k) = Omega(k) = A072047(k). - Lekraj Beedassy, Jul 11 2006

The LCM of any finite subset is in this sequence. - Lekraj Beedassy, Jul 11 2006

This sequence and the Beatty Pi^2/6 sequence (A059535) are "incestuous": the first 20000 terms are bounded within (-9, 14). - Ed Pegg Jr, Jul 22 2008

Let us introduce a function D(n) = sigma_0(n)/2^(alpha(1) + ... + alpha(r)), sigma_0(n) number of divisors of n (A000005), prime factorization of n = p(1)^alpha(1) * ... * p(r)^alpha(r), alpha(1) + ... + alpha(r) is sequence (A001222). Function D(n) splits the set of positive integers into subsets, according to the value of D(n). Squarefree numbers (A005117) has D(n)=1, other numbers are "deviated" from the squarefree ideal and have 0 < D(n) < 1. For D(n)=1/2 we have A048109, for D(n)=3/4 we have A060687. - Ctibor O. Zizka, Sep 21 2008

Numbers k such that gcd(k,k')=1 where k' is the arithmetic derivative (A003415) of k. - Giorgio Balzarotti, Apr 23 2011

Numbers k such that A007913(k) = core(k) = k. - Franz Vrabec, Aug 27 2011

Numbers k such that sqrt(k) cannot be simplified. - Sean Loughran, Sep 04 2011

Indices m where A057918(m)=0, i.e., positive integers m for which there are no integers k in {1,2,...,m-1} such that k*m is a square. - John W. Layman, Sep 08 2011

It appears that these are numbers j such that Product_{k=1..j} (prime(k) mod j) = 0 (see Maple code). - Gary Detlefs, Dec 07 2011. - This is the same claim as Mohammed Bouayoun's Mar 30 2004 comment above. To see why it holds: Primorial numbers, A002110, a subsequence of this sequence, are never divisible by any nonsquarefree number, A013929, and on the other hand, the index of the greatest prime dividing any n is less than n. Cf. A243291. - Antti Karttunen, Jun 03 2014

Conjecture: For each n=2,3,... there are infinitely many integers b > a(n) such that Sum_{k=1..n} a(k)*b^(k-1) is prime, and the smallest such an integer b does not exceed (n+3)*(n+4). - Zhi-Wei Sun, Mar 26 2013

The probability that a random natural number belongs to the sequence is 6/Pi^2, A059956 (see Cesàro reference). - Giorgio Balzarotti, Nov 21 2013

Booker, Hiary, & Keating give a subexponential algorithm for testing membership in this sequence without factoring. - Charles R Greathouse IV, Jan 29 2014

Because in the factorizations into prime numbers these a(n) (n >= 2) have exponents which are either 0 or 1 one could call the a(n) 'numbers with a fermionic prime number decomposition'. The levels are the prime numbers prime(j), j >= 1, and the occupation numbers (exponents) e(j) are 0 or 1 (like in Pauli's exclusion principle). A 'fermionic state' is then denoted by a sequence with entries 0 or 1, where, except for the zero sequence, trailing zeros are omitted. The zero sequence stands for a(1) = 1. For example a(5) = 6 = 2^1*3^1 is denoted by the 'fermionic state' [1, 1], a(7) = 10 by [1, 0, 1]. Compare with 'fermionic partitions' counted in A000009. - Wolfdieter Lang, May 14 2014

From Vladimir Shevelev, Nov 20 2014: (Start)

The following is an Eratosthenes-type sieve for squarefree numbers. For integers > 1:

1) Remove even numbers, except for 2; the minimal non-removed number is 3.

2) Replace multiples of 3 removed in step 1, and remove multiples of 3 except for 3 itself; the minimal non-removed number is 5.

3) Replace multiples of 5 removed as a result of steps 1 and 2, and remove multiples of 5 except for 5 itself; the minimal non-removed number is 6.

4) Replace multiples of 6 removed as a result of steps 1, 2 and 3 and remove multiples of 6 except for 6 itself; the minimal non-removed number is 7.

5) Repeat using the last minimal non-removed number to sieve from the recovered multiples of previous steps.

Proof. We use induction. Suppose that as a result of the algorithm, we have found all squarefree numbers less than n and no other numbers. If n is squarefree, then the number of its proper divisors d > 1 is even (it is 2^k - 2, where k is the number of its prime divisors), and, by the algorithm, it remains in the sequence. Otherwise, n is removed, since the number of its squarefree divisors > 1 is odd (it is 2^k-1).

(End)

The lexicographically least sequence of integers > 1 such that each entry has an even number of proper divisors occurring in the sequence (that's the sieve restated). - Glen Whitney, Aug 30 2015

0 is nonsquarefree because it is divisible by any square. - Jon Perry, Nov 22 2014, edited by M. F. Hasler, Aug 13 2015

The Heinz numbers of partitions with distinct parts. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} prime(j) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] the Heinz number is 2*2*3*7*29 = 2436. The number 30 (= 2*3*5) is in the sequence because it is the Heinz number of the partition [1,2,3]. - Emeric Deutsch, May 21 2015

It is possible for 2 consecutive terms to be even; for example a(258)=422 and a(259)=426. - Thomas Ordowski, Jul 21 2015. [These form a subsequence of A077395 since their product is divisible by 4. - M. F. Hasler, Aug 13 2015]

There are never more than 3 consecutive terms. Runs of 3 terms start at 1, 5, 13, 21, 29, 33, ... (A007675). - Ivan Neretin, Nov 07 2015

a(n) = product of row n in A265668. - Reinhard Zumkeller, Dec 13 2015

Numbers without excess, i.e., numbers k such that A001221(k) = A001222(k). - Juri-Stepan Gerasimov, Sep 05 2016

Numbers k such that b^(phi(k)+1) == b (mod k) for every integer b. - Thomas Ordowski, Oct 09 2016

Boreico shows that the set of square roots of the terms of this sequence is linearly independent over the rationals. - Jason Kimberley, Nov 25 2016 (reference found by Michael Coons).

Numbers k such that A008836(k) = A008683(k). - Enrique Pérez Herrero, Apr 04 2018

The prime zeta function P(s) "has singular points along the real axis for s=1/k where k runs through all positive integers without a square factor". See Wolfram link. - Maleval Francis, Jun 23 2018

Numbers k such that A007947(k) = k. - Kyle Wyonch, Jan 15 2021

The Schnirelmann density of the squarefree numbers is 53/88 (Rogers, 1964). - Amiram Eldar, Mar 12 2021

Comment from Isaac Saffold, Dec 21 2021: (Start)

Numbers k such that all groups of order k have a trivial Frattini subgroup [Dummit and Foote].

Let the group G have order n. If n is squarefree and n > 1, then G is solvable, and thus by Hall's Theorem contains a subgroup H_p of index p for all p | n. Each H_p is maximal in G by order considerations, and the intersection of all the H_p's is trivial. Thus G's Frattini subgroup Phi(G), being the intersection of G's maximal subgroups, must be trivial. If n is not squarefree, the cyclic group of order n has a nontrivial Frattini subgroup. (End)

Numbers for which the squarefree divisors (A206778) and the unitary divisors (A077610) are the same; moreover they are also the set of divisors (A027750). - Bernard Schott, Nov 04 2022

0 = A008683(a(n)) - A008836(a(n)) = A001615(a(n)) - A000203(a(n)). - Torlach Rush, Feb 08 2023

From Robert D. Rosales, May 20 2024: (Start)

Numbers n such that mu(n) != 0, where mu(n) is the Möbius function (A008683).

Solutions to the equation Sum_{d|n} mu(d)*sigma(d) = mu(n)*n, where sigma(n) is the sum of divisors function (A000203). (End)

a(n) is the smallest root of x = 1 + Sum_{k=1..n-1} floor(sqrt(x/a(k))) greater than a(n-1). - Yifan Xie, Jul 10 2024

Number k such that A001414(k) = A008472(k). - Torlach Rush, Apr 14 2025

To elaborate on the formula from Greathouse (2018), the maximum of a(n) - floor(n*Pi^2/6 + sqrt(n)/17) is 10 at indices n = 48715, 48716, 48721, and 48760. The maximum is 11, at the same indices, if floor is taken individually for the two addends and the square root. If the value is rounded instead, the maximum is 9 at 10 indices between 48714 and 48765. - M. F. Hasler, Aug 08 2025

References

• Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 165, p. 53, Ellipses, Paris, 2008.
• David S. Dummit and Richard M. Foote, Abstract algebra. Vol. 1999. Englewood Cliffs, NJ: David S.Prentice Hall, 1991.
• Ivan M. Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 251.
• Michael Pohst and Hans J. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, page 432.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Daniel Forgues, Table of n, a(n) for n = 1..60794 (first 10000 terms from T. D. Noe)
• Zenon B. Batang, Squarefree integers and the abc conjecture, arXiv:2109.10226 [math.GM], 2021.
• Andrew R. Booker, Ghaith A. Hiary and Jon P. Keating, Detecting squarefree numbers, Duke Mathematical Journal, Vol. 164, No. 2 (2015), pp. 235-275; arXiv preprint, arXiv:1304.6937 [math.NT], 2013-2015.
• Iurie Boreico, Linear independence of radicals, The Harvard College Mathematics Review 2(1), 87-92, Spring 2008.
• Ernesto Cesàro, La serie di Lambert in aritmetica assintotica, Rendiconto della Reale Accademia delle Scienze di Napoli, Serie 2, Vol. 7 (1893), pp. 197-204.
• Henri Cohen, Francois Dress, and Mohamed El Marraki, Explicit estimates for summatory functions linked to the Möbius μ-function, Functiones et Approximatio Commentarii Mathematici 37 (2007), part 1, pp. 51-63.
• H. Gent, Letter to N. J. A. Sloane, Nov 27 1975.
• Andrew Granville, ABC means we can count squarefrees, International Mathematical Research Notices 19 (1998), 991-1009.
• Pentti Haukkanen, Mika Mattila, Jorma K. Merikoski and Timo Tossavainen, Can the Arithmetic Derivative be Defined on a Non-Unique Factorization Domain?, Journal of Integer Sequences, Vol. 16 (2013), Article 13.1.2.
• Aaron Krowne, squarefree number, PlanetMath.org.
• Louis Marmet, First occurrences of squarefree gaps and an algorithm for their computation.
• Louis Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012.
• Srinivasa Ramanujan, Irregular numbers, J. Indian Math. Soc., Vol. 5 (1913), pp. 105-106.
• Kenneth Rogers, The Schnirelmann density of the squarefree integers, Proceedings of the American Mathematical Society, Vol. 15, No. 4 (1964), pp. 515-516.
• J. A. Scott, Square-freedom revisited, The Mathematical Gazette, Vol. 90, No. 517 (2006), pp. 112-113.
• Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv preprint arXiv:1511.03860 [math.NT], 2015.
• O. Trifonov, On the Squarefree Problem II, Math. Balkanica, Vol. 3 (1989), Fasc. 3-4.
• Eric Weisstein's World of Mathematics, Squarefree.
• Eric Weisstein's World of Mathematics, Prime Zeta Function.
• Wikipedia, Squarefree integer.
• Index entries for "core" sequences.

Crossrefs

Complement of A013929. Subsequence of A072774 and A209061.

Characteristic function: A008966 (mu(n)^2, where mu = A008683).

Subsequences: A000040, A002110, A235488.

Cf. A001414, A008472, A076259 (first differences), A173143 (partial sums), A000688, A003277, A013928, A020753, A020754, A020755, A030059, A030229, A033197, A034444, A039956, A048672, A053797, A057918, A059956, A071403, A072284, A120992, A133466, A136742, A136743, A160764, A243289, A243347, A243348, A243351, A215366, A046660, A265668, A265675.

Cf. A027750, A077610, A206778.

Subsequences: numbers j such that j*a(k) is squarefree where k > 1: A056911 (k = 2), A261034 (k = 3), A274546 (k = 5), A276378 (k = 6).

Programs

Haskell

a005117 n = a005117_list !! (n-1)
a005117_list = filter ((== 1) . a008966) [1..]
-- Reinhard Zumkeller, Aug 15 2011, May 10 2011

Magma

[ n : n in [1..1000] | IsSquarefree(n) ];

Maple

with(numtheory); a := [ ]; for n from 1 to 200 do if issqrfree(n) then a := [ op(a), n ]; fi; od:
t:= n-> product(ithprime(k),k=1..n): for n from 1 to 113 do if(t(n) mod n = 0) then print(n) fi od; # Gary Detlefs, Dec 07 2011
A005117 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if numtheory[issqrfree](a) then return a; end if; end do: end if; end proc:  # R. J. Mathar, Jan 09 2013

Mathematica

Select[ Range[ 113], SquareFreeQ] (* Robert G. Wilson v, Jan 31 2005 *)
Select[Range[150], Max[Last /@ FactorInteger[ # ]] < 2 &] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006 *)
NextSquareFree[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sf = n + sgn; While[c < Abs[k], While[ ! SquareFreeQ@ sf, If[sgn < 0, sf--, sf++]]; If[ sgn < 0, sf--, sf++]; c++]; sf + If[ sgn < 0, 1, -1]]; NestList[ NextSquareFree, 1, 70] (* Robert G. Wilson v, Apr 18 2014 *)
Select[Range[250], MoebiusMu[#] != 0 &] (* Robert D. Rosales, May 20 2024 *)

PARI

bnd = 1000; L = vector(bnd); j = 1; for (i=1,bnd, if(issquarefree(i),L[j]=i; j=j+1)); L

PARI

{a(n)= local(m,c); if(n<=1,n==1, c=1; m=1; while( cMichael Somos, Apr 29 2005 */

PARI

list(n)=my(v=vectorsmall(n,i,1),u,j); forprime(p=2,sqrtint(n), forstep(i=p^2, n, p^2, v[i]=0)); u=vector(sum(i=1,n,v[i])); for(i=1,n,if(v[i],u[j++]=i)); u \\ Charles R Greathouse IV, Jun 08 2012

PARI

for(n=1, 113, if(core(n)==n, print1(n, ", "))); \\ Arkadiusz Wesolowski, Aug 02 2016

PARI

S(n) = my(s); forsquarefree(k=1,sqrtint(n),s+=n\k[1]^2*moebius(k)); s;
a(n) = my(min=1, max=231, k=0, sc=0); if(n >= 144, min=floor(zeta(2)*n - 5*sqrt(n)); max=ceil(zeta(2)*n + 5*sqrt(n))); while(min <= max, k=(min+max)\2; sc=S(k); if(abs(sc-n) <= sqrtint(n), break); if(sc > n, max=k-1, if(sc < n, min=k+1, break))); while(!issquarefree(k), k-=1); while(sc != n, my(j=1); if(sc > n, j = -1); k += j; sc += j; while(!issquarefree(k), k += j)); k; \\ Daniel Suteu, Jul 07 2022

PARI

first(n)=my(v=vector(n),i); forsquarefree(k=1,if(n<268293,(33*n+30)\20,(n*Pi^2/6+0.058377*sqrt(n))\1), if(i++>n, return(v)); v[i]=k[1]); v \\ Charles R Greathouse IV, Jan 10 2023

PARI

A5117=[1..3]; A005117(n)={if(n>#A5117, my(N=#A5117); A5117=Vec(A5117, max(n+999, N*5\4)); iferr(forsquarefree(k=A5117[N]+1, #A5117*Pi^2\6+sqrtint(#A5117)\17+11, A5117[N++]=k[1]),E,)); A5117[n]} \\ M. F. Hasler, Aug 08 2025

Python

from sympy.ntheory.factor_ import core
def ok(n): return core(n, 2) == n
print(list(filter(ok, range(1, 114)))) # Michael S. Branicky, Jul 31 2021

Python

from itertools import count, islice
from sympy import factorint
def A005117_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(x == 1 for x in factorint(n).values()),count(max(startvalue,1)))
A005117_list = list(islice(A005117_gen(),20)) # Chai Wah Wu, May 09 2022

Python

from math import isqrt
from sympy import mobius
def A005117(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Jul 22 2024

Formula

Limit_{n->oo} a(n)/n = Pi^2/6 (see A013661). - Benoit Cloitre, May 23 2002

Equals A039956 UNION A056911. - R. J. Mathar, May 16 2008

A122840(a(n)) <= 1; A010888(a(n)) < 9. - Reinhard Zumkeller, Mar 30 2010

a(n) = A055229(A062838(n)) and a(n) > A055229(m) for m < A062838(n). - Reinhard Zumkeller, Apr 09 2010

A008477(a(n)) = 1. - Reinhard Zumkeller, Feb 17 2012

A055653(a(n)) = a(n); A055654(a(n)) = 0. - Reinhard Zumkeller, Mar 11 2012

A008966(a(n)) = 1. - Reinhard Zumkeller, May 26 2012

Sum_{n>=1} 1/a(n)^s = zeta(s)/zeta(2*s). - Enrique Pérez Herrero, Jul 07 2012

A056170(a(n)) = 0. - Reinhard Zumkeller, Dec 29 2012

A013928(a(n)+1) = n. - Antti Karttunen, Jun 03 2014

A046660(a(n)) = 0. - Reinhard Zumkeller, Nov 29 2015

Equals {1} UNION A000040 UNION A006881 UNION A007304 UNION A046386 UNION A046387 UNION A067885 UNION A123321 UNION A123322 UNION A115343 ... - R. J. Mathar, Nov 05 2016

|a(n) - n*Pi^2/6| < 0.058377*sqrt(n) for n >= 268293; this result can be derived from Cohen, Dress, & El Marraki, see links. - Charles R Greathouse IV, Jan 18 2018

From Amiram Eldar, Jul 07 2021: (Start)

Sum_{n>=1} (-1)^(a(n)+1)/a(n)^2 = 9/Pi^2.

Sum_{k=1..n} 1/a(k) ~ (6/Pi^2) * log(n).

Sum_{k=1..n} (-1)^(a(k)+1)/a(k) ~ (2/Pi^2) * log(n).

(all from Scott, 2006) (End)