cp's OEIS Frontend

m and m+1 squarefree implies that m*(m+1) is a squarefree oblong number and that m*(m+1)/2 is a squarefree triangular number. - Daniel Forgues, Aug 18 2012

Numbers m such that A002378(m) is squarefree. - Thomas Ordowski, Sep 01 2015

For n >= 1 define an n X n (0, 1) matrix A by A[i, j] = 1 if gcd(i, j) = 1, A[i, j] = 0 if gcd(i, j) > 1 for 1 <= i,j <= n . The rank of A is a(n + 1). Asymptotic expression for a(n) is a(n) ~ n * 6 / Pi^2. - Sharon Sela (sharonsela(AT)hotmail.com), May 06 2002

a(n) = Sum_{k=1..n-1} A008966(k). - Reinhard Zumkeller, Jul 05 2010

For all n >= 1, a(n)/n >= a(176)/176 = 53/88, and the equality occurs only for n=176 (see K. Rogers link). - Michel Marcus, Dec 16 2012 [Thus the Schnirelmann density of the squarefree numbers is 53/88. - Charles R Greathouse IV, Feb 02 2016]

Cohen, Dress, & El Marraki prove that |a(n) - 6n/Pi^2| < 0.02767*sqrt(n) for n >= 438653. - Charles R Greathouse IV, Feb 02 2016

This is a multiplicative self-inverse permutation of the integers.

A225547 gives the fixed points.

From Antti Karttunen and Peter Munn, Feb 02 2020: (Start)

This sequence operates on the Fermi-Dirac factors of a number. As arranged in array form, in A329050, this sequence reflects these factors about the main diagonal of the array, substituting A329050[j,i] for A329050[i,j], and this results in many relationships including significant homomorphisms.

This sequence provides a relationship between the operations of squaring and prime shift (A003961) because each successive column of the A329050 array is the square of the previous column, and each successive row is the prime shift of the previous row.

A329050 gives examples of how significant sets of numbers can be formed by choosing their factors in relation to rows and/or columns. This sequence therefore maps equivalent derived sets by exchanging rows and columns. Thus odd numbers are exchanged for squares, squarefree numbers for powers of 2 etc.

Alternative construction: For n > 1, form a vector v of length A299090(n), where each element v[i] for i=1..A299090(n) is a product of those distinct prime factors p(i) of n whose exponent e(i) has the bit (i-1) "on", or 1 (as an empty product) if no such exponents are present. a(n) is then Product_{i=1..A299090(n)} A000040(i)^A048675(v[i]). Note that because each element of vector v is squarefree, it means that each exponent A048675(v[i]) present in the product is a "submask" (not all necessarily proper) of the binary string A087207(n).

This permutation effects the following mappings:

A000035(a(n)) = A010052(n), A010052(a(n)) = A000035(n). [Odd numbers <-> Squares]

A008966(a(n)) = A209229(n), A209229(a(n)) = A008966(n). [Squarefree numbers <-> Powers of 2]

(End)

From Antti Karttunen, Jul 08 2020: (Start)

Moreover, we see also that this sequence maps between A016825 (Numbers of the form 4k+2) and A001105 (2*squares) as well as between A008586 (Multiples of 4) and A028983 (Numbers with even sum of the divisors).

(End)

Except for a(2)=2, all the terms in the sequence are odd. This is because of the existence of a non-cyclic dihedral group of order 2n for each n>1. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 09 2001

Also gcd(n, A051953(n)) = 1. - Labos Elemer

n such that x^n == 1 (mod n) has no solution 2 <= x <= n. - Benoit Cloitre, May 10 2002

There is only one group (the cyclic group of order n) whose order is n. - Gerard P. Michon, Jan 08 2008 [This is a 1947 result of Tibor Szele. - Charles R Greathouse IV, Nov 23 2011]

Any divisor of a Carmichael number (A002997) must be odd and cyclic. Conversely, G. P. Michon conjectured (c. 1980) that any odd cyclic number has at least one Carmichael multiple (if the conjecture is true, each of them has infinitely many such multiples). In 2007, Michon & Crump produced explicit Carmichael multiples of all odd cyclic numbers below 10000 (see link, cf. A253595). - Gerard P. Michon, Jan 08 2008

Numbers n such that phi(n)^phi(n) == 1 (mod n). - Michel Lagneau, Nov 18 2012

Contains A000040, and all members of A006094 except 6. - Robert Israel, Jul 08 2015

Number m such that n^n == r (mod m) is solvable for any r. - David W. Wilson, Oct 01 2015

Numbers m such that A074792(m) = m + 1. - Thomas Ordowski, Jul 16 2017

Squarefree terms of A056867 (see McCarthy link p. 592 and similar comment with "cubefree" in A051532). - Bernard Schott, Mar 24 2022

From Daniel Forgues, May 27 2009: (Start)

For any prime p, there are as many squarefree numbers having p as a factor as squarefree numbers not having p as a factor amongst all the squarefree numbers (one-to-one correspondence, both cardinality aleph_0).

E.g. there are as many even squarefree numbers as there are odd squarefree numbers.

For any prime p, the density of squarefree numbers having p as a factor is 1/p of the density of squarefree numbers not having p as a factor.

E.g. the density of even squarefree numbers is 1/p = 1/2 of the density of odd squarefree numbers (which means that 1/(p + 1) = 1/3 of the squarefree numbers are even and p/(p + 1) = 2/3 are odd). As a consequence the n-th even squarefree number is very nearly p = 2 times the n-th odd squarefree number (which means that the n-th even squarefree number is very nearly (p + 1) = 3 times the n-th squarefree number while the n-th odd squarefree number is very nearly (p + 1)/p = 3/2 the n-th squarefree number).

For any prime p, the n-th squarefree number not divisible by p is: n * (1 + 1/p) * zeta(2) + O(n^(1/2)) = n * (1 + 1/p) * (Pi^2 / 6) + O(n^(1/2)) (End)

A prime factor of n is unitary iff its exponent is 1 in the prime factorization of n. (Of course for any prime p, GCD(p, n/p) is either 1 or p. For a unitary prime factor it must be 1.)

Number of squared primes dividing n. - Reinhard Zumkeller, May 18 2002

a(A005117(n)) = 0; a(A013929(n)) > 0; a(A190641(n)) = 1. - Reinhard Zumkeller, Dec 29 2012

First differences of A013940. - Jason Kimberley, Feb 01 2017

Number of exponents larger than 1 in the prime factorization of n. - Antti Karttunen, Nov 28 2017

Euler transform of the absolute values of A008683. - Tilman Neumann, Dec 13 2008

Euler transform of A008966. - Vaclav Kotesovec, Mar 31 2018

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

The number of unordered factorizations of n into 1 and exponentially odd prime powers, i.e., p^e where p is a prime and e is odd (A246551). - Amiram Eldar, Jun 12 2025

Nonprime squarefree numbers.

Except for 1, composite n such that the squarefree part of n is greater than phi(n). - Benoit Cloitre, Apr 06 2002

Numbers having only squarefree exponents in their canonical prime factorization.

According to the formula of Theorem 3 [Toth], the density of the exponentially squarefree numbers is 0.9559230158619... (A262276). - Peter J. C. Moses and Vladimir Shevelev, Sep 10 2015

From Vladimir Shevelev, Sep 24 2015: (Start)

A generalization. Let S be a finite or infinite increasing integer sequence s=s(n), s(0)=0.

Let us call a positive number N an exponentially S-number, if all exponents in its prime power factorization are in the sequence S.

Let {u(n)} be the characteristic function of S. Then, for the density h=h(S) of the exponentially S-numbers, we have the representations

h(S) = Product_{prime p} Sum_{j in S} (p-1)/p^(j+1) = Product_{p} (1 + Sum_{j>=1} (u(j) - u(j-1))/p^j). In particular, if S = {0,1}, then the exponentially S-numbers are squarefree numbers; if S consists of 0 and {2^k}A138302%20(see%20%5BShevelev%5D,%202007);%20if%20S%20consists%20of%200%20and%20squarefree%20numbers,%20then%20u(n)=%7Cmu(n)%7C,%20where%20mu(n)%20is%20the%20M%C3%B6bius%20function%20(A008683),%20we%20obtain%20the%20density%20h%20of%20the%20exponentially%20squarefree%20numbers%20(cf.%20Toth's%20link,%20Theorem%203);%20the%20calculation%20of%20h%20with%20a%20very%20high%20degree%20of%20accuracy%20belongs%20to%20_Juan%20Arias-de-Reyna">{k>=0}, then the exponentially S-numbers form A138302 (see [Shevelev], 2007); if S consists of 0 and squarefree numbers, then u(n)=|mu(n)|, where mu(n) is the Möbius function (A008683), we obtain the density h of the exponentially squarefree numbers (cf. Toth's link, Theorem 3); the calculation of h with a very high degree of accuracy belongs to _Juan Arias-de-Reyna (A262276). Note that if S contains 1, then h(S) >= 1/zeta(2) = 6/Pi^2; otherwise h(S) = 0. Indeed, in the latter case, the density of the sequence of exponentially S-numbers does not exceed the density of A001694, which equals 0. (End)

The term "exponentially squarefree number" was apparently coined by Subbarao (1972). - Amiram Eldar, May 28 2025