A013929 Numbers that are not squarefree. Numbers that are divisible by a square greater than 1. The complement of A005117.

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160

Offset: 1

Author

Henri Lifchitz

Keywords

• nonn
• easy

Comments

Sometimes misnamed squareful numbers, but officially those are given by A001694.

This is different from the sequence of numbers k such that A007913(k) < phi(k). The two sequences differ at the values: 420, 660, 780, 840, 1320, 1560, 4620, 5460, 7140, ..., which is essentially A070237. - Ant King, Dec 16 2005

Numbers k such that Sum_{d|k} (d/phi(d))*mu(k/d) = 0. - Benoit Cloitre, Apr 28 2002

Also, k with at least one x < k such that A007913(x) = A007913(k). - Benoit Cloitre, Apr 28 2002

Numbers k for which there exists a partition into two parts p and q such that p + q = k and p*q is a multiple of k. - Amarnath Murthy, May 30 2003

Numbers k such that there is a solution 0 < x < k to x^2 == 0 (mod k). - Franz Vrabec, Aug 13 2005

Numbers k such that moebius(k) = 0.

a(n) = k such that phi(k)/k = phi(m)/m for some m < k. - Artur Jasinski, Nov 05 2008

Appears to be numbers such that when a column with index equal to a(n) in A051731 is deleted, there is no impact on the result in the first column of A054525. - Mats Granvik, Feb 06 2009

Numbers k such that the number of prime divisors of (k+1) is less than the number of nonprime divisors of (k+1). - Juri-Stepan Gerasimov, Nov 10 2009

Orders for which at least one non-cyclic finite abelian group exists: A000688(a(n)) > 1. This follows from the fact that not all exponents in the prime factorization of a(n) are 1 (moebius(a(n)) = 0). The number of such groups of order a(n) is A192005(n) = A000688(a(n)) - 1. - Wolfdieter Lang, Jul 29 2011

Subsequence of A193166; A192280(a(n)) = 0. - Reinhard Zumkeller, Aug 26 2011

It appears that terms are the numbers m such that Product_{k=1..m} (prime(k) mod m) <> 0. See Maple code. - Gary Detlefs, Dec 07 2011

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

A057918(a(n)) > 0. - Reinhard Zumkeller, Mar 27 2012

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

Numbers k such that A001221(k) != A001222(k). - Felix Fröhlich, Aug 13 2014

Numbers k such that A001222(k) > A001221(k), since in this case at least one prime factor of k occurs more than once, which implies that k is divisible by at least one perfect square > 1. - Carlos Eduardo Olivieri, Aug 02 2015

Lexicographically least sequence such that each term has a positive even number of proper divisors not occurring in the sequence, cf. the sieve characterization of A005117. - Glen Whitney, Aug 30 2015

There are arbitrarily long runs of consecutive terms. Record runs start at 4, 8, 48, 242, ... (A045882). - Ivan Neretin, Nov 07 2015

A number k is a term if 0 < min(A000010(k) + A023900(k), A000010(k) - A023900(k)). - Torlach Rush, Feb 22 2018

Every squareful number > 1 is nonsquarefree, but the converse is false and the nonsquarefree numbers that are not squareful (see first comment) are in A332785. - Bernard Schott, Apr 11 2021

Integers m where at least one k < m exists such that m divides k^m. - Richard R. Forberg, Jul 31 2021

Consider the Diophantine equation S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = z, when x and y are both positive integers with y | x. Then, there is a solution (x,y) iff z is a term of this sequence; in this case, if x = K*y, then z = S(K*y,y) = K*(y+1)^2 (see A351381, link and references Perelman); example: S(12,4) = 75 = a(28). The number of solutions for S(x,y) = a(n) is A353282(n). - Bernard Schott, Mar 29 2022

For each positive integer m, the number of unitary divisors of m = the number of squarefree divisors of m (see A034444); but only for the terms of this sequence does the set of unitary divisors differ from the set of squarefree divisors. Example: the set of unitary divisors of 20 is {1, 4, 5, 20}, while the set of squarefree divisors of 20 is {1, 2, 5, 10}. - Bernard Schott, Oct 15 2022

Examples

			For the terms up to 20, we compute the squares of primes up to floor(sqrt(20)) = 4. Those squares are 4 and 9. For every such square s, put the terms s*k^2 for k = 1 to floor(20 / s). This gives after sorting and removing duplicates the list 4, 8, 9, 12, 16, 18, 20. - _David A. Corneth_, Oct 25 2017
		

References

• I. Perelman, L'Algèbre récréative, Deux nombres et quatre opérations, Editions en langues étrangères, Moscou, 1959, pp. 101-102.
• Ya. I. Perelman, Algebra can be fun, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.

Links

• David A. Corneth, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
• H. Gent, Letter to N. J. A. Sloane, Nov 27 1975.
• Louis Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv:1210.3829 [math.NT], 2012.
• Ya. I. Perelman, Algebra Can Be Fun, Chapter IV, Diophantine Equations, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.
• Srinivasa Ramanujan, Irregular numbers, J. Indian Math. Soc. 5 (1913) 105-106.
• Eric Weisstein's World of Mathematics, Smarandache Near-to-Primorial Function, Squarefree, Squareful, Moebius Function.

Crossrefs

Complement of A005117. Subsequences: A130897, A190641, A332785.

Cf. A001694, A008683, A034444, A038109, A351381, A353282.

Partitions into: A114374, A256012.

Programs

Haskell

a013929 n = a013929_list !! (n-1)
a013929_list = filter ((== 0) . a008966) [1..]
-- Reinhard Zumkeller, Apr 22 2012

Magma

[ n : n in [1..1000] | not IsSquarefree(n) ];

Maple

a := n -> `if`(numtheory[mobius](n)=0,n,NULL); seq(a(i),i=1..160); # Peter Luschny, May 04 2009
t:= n-> product(ithprime(k),k=1..n): for n from 1 to 160 do (if t(n) mod n <>0) then print(n) fi od; # Gary Detlefs, Dec 07 2011
with(NumberTheory): isQuadrateful := n -> irem(Radical(n), n) <> 0:
select(isQuadrateful, [`$`(1..160)]);  # Peter Luschny, Jul 12 2022

Mathematica

Union[ Flatten[ Table[ n i^2, {i, 2, 20}, {n, 1, 400/i^2} ] ] ]
Select[ Range[2, 160], (Union[Last /@ FactorInteger[ # ]][[ -1]] > 1) == True &] (* Robert G. Wilson v, Oct 11 2005 *)
Cases[Range[160], n_ /; !SquareFreeQ[n]] (* Jean-François Alcover, Mar 21 2011 *)
Select[Range@160, ! SquareFreeQ[#] &] (* Robert G. Wilson v, Jul 21 2012 *)
Select[Range@160, PrimeOmega[#] > PrimeNu[#] &] (* Carlos Eduardo Olivieri, Aug 02 2015 *)
Select[Range[200], MoebiusMu[#] == 0 &] (* Alonso del Arte, Nov 07 2015 *)

PARI

{a(n)= local(m,c); if(n<=1,4*(n==1), c=1; m=4; while( cMichael Somos, Apr 29 2005 */

PARI

for(n=1, 1e3, if(omega(n)!=bigomega(n), print1(n, ", "))) \\ Felix Fröhlich, Aug 13 2014

PARI

upto(n)=my(res = List()); forprime(p = 2, sqrtint(n), for(k = 1, n \ p^2, listput(res, k * p^2))); listsort(res, 1); res \\ David A. Corneth, Oct 25 2017

Python

from sympy.ntheory.factor_ import core
def ok(n): return core(n, 2) != n
print(list(filter(ok, range(1, 161)))) # Michael S. Branicky, Apr 08 2021

Python

from math import isqrt
from sympy import mobius
def A013929(n):
    def f(x): return n+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 20 2024

Formula

A008966(a(n)) = 0. - Reinhard Zumkeller, Apr 22 2012

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

a(n) ~ n/k, where k = 1 - 1/zeta(2) = 1 - 6/Pi^2 = A229099. - Charles R Greathouse IV, Sep 13 2013

A001222(a(n)) > A001221(a(n)). - Carlos Eduardo Olivieri, Aug 02 2015

phi(a(n)) > A003958(a(n)). - Juri-Stepan Gerasimov, Apr 09 2019

Extensions

More terms from Erich Friedman

More terms from Franz Vrabec, Aug 13 2005