A229099 -id:A229099 - cp's OEIS frontend

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

Henri Lifchitz

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

			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

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.

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.

Complement of A005117. Subsequences: A130897, A190641, A332785.
Cf. A001694, A008683, A034444, A038109, A351381, A353282.
Partitions into: A114374, A256012.

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

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

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

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 *)

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

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

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

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

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

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

More terms from Erich Friedman

More terms from Franz Vrabec, Aug 13 2005

A078147 First differences of sequence of nonsquarefree numbers, A013929.

Original entry on oeis.org

4, 1, 3, 4, 2, 2, 4, 1, 2, 1, 4, 4, 4, 4, 1, 3, 1, 1, 2, 2, 2, 4, 3, 1, 4, 4, 3, 1, 4, 1, 3, 4, 2, 2, 4, 2, 1, 1, 4, 4, 4, 4, 1, 3, 1, 3, 1, 1, 2, 4, 3, 1, 4, 4, 3, 1, 2, 2, 1, 3, 4, 2, 2, 4, 1, 2, 1, 3, 1, 4, 4, 4, 1, 3, 4, 2, 2, 4, 3, 1, 4, 4, 4, 4, 1, 3, 4, 2, 2, 4, 2, 1, 1, 1, 3, 2, 2, 4, 4, 1, 3, 4, 2, 2, 3

Offset: 1

Labos Elemer, Nov 26 2002

Run lengths in A132345, apart from initial run of zeros. - Reinhard Zumkeller, Apr 22 2012

The asymptotic density of the occurrences of 1 in this sequence is density(A068781)/density(A013929) = (1 - 2 * A059956 + A065474)/A229099 = 0.272347... - Amiram Eldar, Mar 09 2021

			a(1) = 4 = 8 - 4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A068781, A013929, A132345.

Cf. A059956, A065474, A229099.

a078147 n = a078147_list !! (n-1)
a078147_list = zipWith (-) (tail a013929_list) a013929_list
-- Reinhard Zumkeller, Apr 22 2012

t=Flatten[Position[Table[MoebiusMu[w], {w, 1, 1000}], 0]]; t1=Delete[RotateLeft[t]-t, -1]
Differences[Select[Range[300],!SquareFreeQ[#]&]] (* Harvey P. Dale, May 07 2012 *)

lista(nn) = {my(prec=0); for (n=1, nn, if (!issquarefree(n), if (prec, print1(n-prec, ", ")); prec = n;););} \\ Michel Marcus, Mar 26 2020

from math import isqrt
from sympy import mobius, factorint
def A078147(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 next(i for i in range(1,5) if any(d>1 for d in factorint(m+i).values())) # Chai Wah Wu, Sep 10 2024

a(n) = A013929(n+1) - A013929(n).
a(n) = 1, 2, 3 or 4 since n = 4*k is always nonsquarefree.
Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = Pi^2/(Pi^2-6) = 2.550546... - Amiram Eldar, Oct 21 2020

Offset fixed by Reinhard Zumkeller, Apr 22 2012

A048105 Number of non-unitary divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 4, 1, 0, 2, 2, 0, 0, 0, 4, 0, 0, 0, 5, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 6, 1, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 4, 0, 0, 2, 5, 0, 0, 0, 2, 0, 0, 0, 8, 0, 0, 2, 2, 0, 0, 0, 6, 3, 0, 0, 4, 0, 0, 0, 4, 0, 4, 0, 2, 0, 0, 0, 8, 0, 2, 2, 5, 0, 0, 0, 4, 0

Offset: 1

Labos Elemer

nonn

Number of zeros in row n of table A225817. - Reinhard Zumkeller, Jul 30 2013

			Example 1: If n is squarefree (A005117) then a(n)=0 since all divisors are unitary.
Example 2: n=12, d(n)=6, ud(n)=4, nud(12)=d(12)-ud(12)=2; from {1,2,3,4,6,12} {1,3,4,12} are unitary while {2,6} are not unitary divisors.
Example 3: n=p^k, a true prime power, d(n)=k+1, u(d)=2^r(x)=2, so nud(n)=d(p^k)-2=k+1 i.e., it can be arbitrarily large.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A007875, A056170, A225817, A034444.

Cf. A001620, A229099, A306016.

a048105 n = length [d | d <- [1..n], mod n d == 0, gcd d (n `div` d) > 1]
-- Reinhard Zumkeller, Aug 17 2011

with(NumberTheory):
seq(SumOfDivisors(n, 0) - 2^NumberOfPrimeFactors(n, 'distinct'), n = 1..105);
# Peter Luschny, Jul 27 2023

Table[DivisorSigma[0, n] - 2^PrimeNu[n], {n, 1, 50}] (* Geoffrey Critzer, Dec 10 2014 *)

a(n)=my(f=factor(n)[,2]); prod(i=1,#f,f[i]+1)-2^#f \\ Charles R Greathouse IV, Sep 18 2015

from math import prod
from sympy import factorint
def A048105(n): return -(1<Chai Wah Wu, Aug 12 2024

a(n) = Sigma(0, n) - 2^r(n), where r() = A001221, the number of distinct primes dividing n.
From Reinhard Zumkeller, Jul 30 2013: (Start)
a(n) = A000005(n) - A034444(n).
For n > 1: a(n) = A000005(n) - 2 * A007875(n). (End)
Dirichlet g.f.: zeta(s)^2 - zeta(s)^2/zeta(2*s). - Geoffrey Critzer, Dec 10 2014
G.f.: Sum_{k>=1} (1 - mu(k)^2)*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 21 2017
Sum_{k=1..n} a(k) ~ (1-6/Pi^2)*n*log(n) + ((1-6/Pi^2)*(2*gamma-1)+(72*zeta'(2)/Pi^4))*n , where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2022

A057627 Number of nonsquarefree numbers not exceeding n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 7, 7, 7, 7, 8, 9, 9, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 16, 16, 16, 17, 18, 19, 19, 20, 20, 21, 21, 22, 22, 22, 22, 23, 23, 23, 24, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 28, 29, 29, 29

Offset: 1

Labos Elemer, Oct 10 2000

Number of integers k in A013929 in the range 1 <= k <= n.
This sequence is different from A013940, albeit the first 35 terms are identical.
Asymptotic to k*n where k = 1 - 1/zeta(2) = 1 - 6/Pi^2 = A229099. - Daniel Forgues, Jan 28 2011
This sequence is the sequence of partial sums of A107078 (not of A056170). - Jason Kimberley, Feb 01 2017
Number of partitions of 2n into two parts with the smallest part nonsquarefree. - Wesley Ivan Hurt, Oct 25 2017

			a(36)=13 because 13 nonsquarefree numbers exist which do not exceed 36:{4,8,9,12,16,18,20,24,25,27,28,32,36}.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005117, A008683, A013929, A013940, A013928, A002321, A028442, A107078, A229099, A294242.

N:= 1000: # to get terms up to a(N)
B:= Array(1..N, numtheory:-issqrfree):
C:= map(`if`,B,0,1):
A:= map(round,Statistics:-CumulativeSum(C)):
seq(A[n],n=1..N); # Robert Israel, Jun 03 2014

Accumulate[Table[If[SquareFreeQ[n],0,1],{n,80}]] (* Harvey P. Dale, Jun 04 2014 *)

a(n) = my(s=0); forsquarefree(k=1, sqrtint(n), s += (-1)^(#k[2]~) * (n\k[1]^2)); n - s; \\ Charles R Greathouse IV, May 18 2015; corrected by Daniel Suteu, May 11 2023

from math import isqrt
from sympy import mobius
def A057627(n): return n-sum(mobius(k)*(n//k**2) for k in range(1,isqrt(n)+1)) # Chai Wah Wu, May 10 2024

(define (A057627 n) (- n (A013928 (+ n 1))))

a(n) = n - A013928(n+1) = n - Sum_{k=1..n} mu(k)^2.

G.f.: Sum_{k>=1} (1 - mu(k)^2)*x^k/(1 - x). - Ilya Gutkovskiy, Apr 17 2017

Offset and formula corrected by Antti Karttunen, Jun 03 2014

A332785 Nonsquarefree numbers that are not squareful.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 204, 207, 208, 212, 220, 224

Offset: 1

Bernard Schott, Feb 24 2020

Sometimes nonsquarefree numbers are misnamed squareful numbers (see 1st comment of A013929). Indeed, every squareful number > 1 is nonsquarefree, but the converse is false. This sequence = A013929 \ A001694 and consists of these counterexamples.
This sequence is not a duplicate: the first 16 terms (<= 68) are the same first 16 terms of A059404, A323055, A242416 and A303946, then 72 is the 17th term of these 4 sequences. Also, the first 37 terms (<= 140) are the same first 37 terms of A317616 then 144 is the 38th term of this last sequence.
From Amiram Eldar, Sep 17 2023: (Start)
Called "hybrid numbers" by Jakimczuk (2019).
These numbers have a unique representation as a product of two numbers > 1, one is squarefree (A005117) and the other is powerful (A001694).
Equivalently, numbers k such that A055231(k) > 1 and A057521(k) > 1.
Equivalently, numbers that have in their prime factorization at least one exponent that is equal to 1 and at least one exponent that is larger than 1.
The asymptotic density of this sequence is 1 - 1/zeta(2) (A229099). (End)

			18 = 2 * 3^2 is nonsquarefree as it is divisible by the square 3^2, but it is not squareful because 2 divides 18 but 2^2 does not divide 18, hence 18 is a term.
72 = 2^3 * 3^2 is nonsquarefree as it is divisible by the square 3^2, but it is also squareful because primes 2 and 3 divide 72, and 2^2 and 3^2 divide also 72, so 72 is not a term.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Powerful Numbers Multiple of a Set of Primes and Hybrid Numbers, 2019.

Cf. A005117 (squarefree), A013929 (nonsquarefree), A001694 (squareful), A052485 (not squareful).

Cf. A059404, A126706, A229099, A242416, A286708, A303946, A317616, A323055 (first terms are the same).

filter:= proc(n) local F;
 F:= ifactors(n)[2][..,2];
 max(F) > 1 and min(F) = 1
end proc:
select(filter, [$1..1000]); # Robert Israel, Sep 15 2024

Select[Range[225], Max[(e = FactorInteger[#][[;;,2]])] > 1 && Min[e] == 1 &] (* Amiram Eldar, Feb 24 2020 *)

isok(m) = !issquarefree(m) && !ispowerful(m); \\ Michel Marcus, Feb 24 2020

from math import isqrt
from sympy import mobius, integer_nthroot
def A332785(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l, j = n-1+squarefreepi(integer_nthroot(x,3)[0])+squarefreepi(x), 0, isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c += j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c-l
    return bisection(f,n,n) # Chai Wah Wu, Sep 14 2024

This sequence is A126706 \ A286708.

Sum_{n>=1} 1/a(n)^s = 1 + zeta(s) - zeta(s)/zeta(2*s) - zeta(2*s)*zeta(3*s)/zeta(6*s), s > 1. - Amiram Eldar, Sep 17 2023

A107078 Whether n has non-unitary prime divisors.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0

Offset: 1

Paul Barry, May 10 2005

Also the characteristic function of the numbers that are not squarefree: A013929. - Enrique Pérez Herrero, Jul 08 2012

The sequence of partial sums of this sequence is A057627. - Jason Kimberley, Feb 01 2017

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Index entries for characteristic functions.

Cf. A087049. - R. J. Mathar, Aug 24 2008

Cf. A013929, A008683, A008966, A107078, A229099.

seq(1 - abs(numtheory:-mobius(n)), n = 1..101); # Peter Luschny, Jul 27 2023

Table[1-MoebiusMu[n]^2,{n,1,100}] (* Enrique Pérez Herrero, Jul 08 2012 *)

from sympy import mobius
def A107078(n): return int(not mobius(n)) # Chai Wah Wu, Dec 05 2024

a(n) = 1 if A056170(n)>0, 0 otherwise.
a(n) = A107079(n) - A013928(n+1).
a(n) = 1 - A008966(n). - Reinhard Zumkeller, Oct 03 2008
a(n) = Sum_{k=0..n-1} (mu(n-k-1) mod 2) - Sum_{k=0..n-1} (mu(n-k) mod 2).
a(n) = abs(mu(n) - (-1)^omega(n)) = (mu(n) - (-1)^omega(n))^2 = abs(A008683(n) - (-1)^A001221(n)). - Enrique Pérez Herrero, Apr 28 2012
a(n) = 1 - mu(n)^2. - Enrique Pérez Herrero, Jul 08 2012
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 - 6/Pi^2 (A229099). - Amiram Eldar, Jul 24 2022

A007424 a(n) = 1 if n is squarefree, otherwise 2.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.

Cf. A005117, A008966, A107078, A229099.

Cf. A007948, A051903, A130130.

Table[If[SquareFreeQ[n],1,2],{n,100}] (* Harvey P. Dale, Jul 09 2014 *)

MuPAD
```
func(2-abs(numlib::moebius(n)), n):
```

A007424(n) = (2-issquarefree(n)); \\ Antti Karttunen, Nov 18 2017

a(n) = 2 - A008966(n). - Antti Karttunen, Nov 18 2017
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 - 6/Pi^2 = 1 + A229099. - Amiram Eldar, Sep 26 2022
a(n) = A051903(A007948(n)) = A130130(A051903(n)) for n >= 2. - Amiram Eldar, May 07 2024

A053650 Cototient function of n^2.

Original entry on oeis.org

0, 2, 3, 8, 5, 24, 7, 32, 27, 60, 11, 96, 13, 112, 105, 128, 17, 216, 19, 240, 189, 264, 23, 384, 125, 364, 243, 448, 29, 660, 31, 512, 429, 612, 385, 864, 37, 760, 585, 960, 41, 1260, 43, 1056, 945, 1104, 47, 1536, 343, 1500, 969, 1456, 53, 1944, 825, 1792, 1197

Offset: 1

Labos Elemer, Feb 18 2000

Seems to be invertible like n*Phi(n). Compare with A002618, A038040.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A038040.

Cf. A001248, A002618, A053650, A053192, A053193, A036689, A229099.

List([1..60], n-> n*(n- Phi(n)) ); # G. C. Greubel, May 18 2019

a053650 = a051953 . a000290  -- Reinhard Zumkeller, Jan 21 2014

[n*(n-EulerPhi(n)): n in [1..60]]; // Vincenzo Librandi, Jul 27 2016

Table[n(n-EulerPhi[n]), {n, 60}] (* Michael De Vlieger, Jul 26 2016 *)

a(n) = n^2 - eulerphi(n^2) \\ Michel Marcus, Jul 27 2013

[n*(n - euler_phi(n)) for n in (1..60)] # G. C. Greubel, May 18 2019

a(n) = n*(n - phi(n)) = n^2 - n*phi(n) = Cototient(n^2) = A051953(A000290(n)).
a(n) = n^2 - A002618(n).
For p prime, Cototient(p)=1 and a(p)=p.
a(n) = n*cototient(n) = n*A051953(n). - Omar E. Pol, Nov 22 2012
Dirichlet g.f.: zeta(s-2)*(1 - 1/zeta(s-1)). - Ilya Gutkovskiy, Jul 26 2016
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 1 - 6/Pi^2 (A229099). - Amiram Eldar, Dec 15 2023

A275699 Excess of numbers that are not squarefree.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 1, 3, 1, 1, 3, 3, 1, 2, 1, 1, 4, 1, 1, 2, 2, 3, 3, 1, 1, 2, 1, 1, 2, 1, 6, 1, 2, 2, 1, 4, 1, 1, 1, 2, 1, 1, 4, 3, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 2, 5, 2, 1, 3, 1, 1, 3, 1, 4, 1, 4, 2, 1

Offset: 1

Felix Fröhlich, Aug 05 2016

nonn

The "excess" of a number is the number of prime divisors with multiplicity (the Omega function, A001222) minus the number of distinct prime divisors (the omega function, A001221). A046660(n) gives the excess of n.

Since squarefree numbers have no excess, this sequence is essentially A046660 with the 0's removed.

			Since 16 = 2^4, 16 has four prime divisors, but only one distinct divisor. Hence Omega(16) - omega(16) = 4 - 1 = 3. As 16 is the fifth number that is not squarefree, its corresponding 3 is a(5) in this sequence.
17 is prime and thus has no excess and no corresponding term in this sequence.
18 = 2 * 3^2, Omega(18) - omega(18) = 3 - 2 = 1, thus a(6) = 1.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A001221, A001222, A013929, A046660, A136141, A229099.

DeleteCases[Table[PrimeOmega[n] - PrimeNu[n], {n, 200}], 0] (* Alonso del Arte, Aug 05 2016 *)

for(n=1, 200, if(bigomega(n)!=omega(n), print1(bigomega(n)-omega(n), ", ")))

a(n) = A046660(A013929(n)).

Asymptotic mean: lim_{m->oo} (1/m) Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p*(p-1)) / (1-6/Pi^2) = A136141/A229099 = 1.9719717... - Amiram Eldar, Feb 10 2021

cp's OEIS Frontend

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

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A078147 First differences of sequence of nonsquarefree numbers, A013929.

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A048105 Number of non-unitary divisors of n.

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A057627 Number of nonsquarefree numbers not exceeding n.

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A332785 Nonsquarefree numbers that are not squareful.

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