A179214 -id:A179214 - cp's OEIS frontend

A008966 a(n) = 1 if n is squarefree, otherwise 0.

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

Offset: 1

N. J. A. Sloane

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 infinite lower triangular matrix with A008966 on the main diagonal and the rest zeros is the square of triangle A143255. - Gary W. Adamson, Aug 02 2008

Daniel Forgues, Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Moebius Function
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n

Cf. A005117, A008836 (Dirichlet inverse), A013928 (partial sums).
Cf. A179211, A179212, A179213, A179214, A179215.
Cf. A020639, A073576, A087188.
Cf. A124010, A212793, A085357, A156552.
Parity of A002033.
Cf. A047999, A036987
Cf. A082020 (Dgf at s=2), A157289 (Dgf at s=3), A157290 (Dgf at s=4).

a008966 = abs . a008683
-- Reinhard Zumkeller, Dec 13 2015, Dec 15 2014, May 27 2012, Jan 25 2012

Magma
```
[ Abs(MoebiusMu(n)) : n in [1..100]];
```

A008966 := proc(n) if numtheory[issqrfree](n) then 1 ; else 0 ; end if; end proc: # R. J. Mathar, Mar 14 2011

A008966[n_] := Abs[MoebiusMu[n]]; Table[A008966[n], {n, 100}] (* Enrique Pérez Herrero, Apr 15 2010 *)
Table[If[SquareFreeQ[n],1,0],{n,100}] (* or *) Boole[SquareFreeQ/@ Range[ 100]] (* Harvey P. Dale, Feb 28 2015 *)

MuPAD
```
func(abs(numlib::moebius(n)), n):
```
PARI
```
a(n)=if(n<1,0,direuler(p=2,n,1+X))[n]
```

a(n)=issquarefree(n) \\ Michel Marcus, Feb 22 2015

from sympy import factorint
def A008966(n): return int(max(factorint(n).values(),default=1)==1) # Chai Wah Wu, Apr 05 2023

Dirichlet g.f.: zeta(s)/zeta(2s).
a(n) = abs(mu(n)), where mu is the Moebius function (A008683).
a(n) = 0^(bigomega(n) - omega(n)), where bigomega(n) and omega(n) are the numbers of prime factors of n with and without repetition (A001222, A001221, A046660). - Reinhard Zumkeller, Apr 05 2003
Multiplicative with p^e -> 0^(e - 1), p prime and e > 0. - Reinhard Zumkeller, Jul 15 2003
a(n) = 0^(A046951(n) - 1). - Reinhard Zumkeller, May 20 2007
a(n) = 1 - A107078(n). - Reinhard Zumkeller, Oct 03 2008
a(n) = floor(rad(n)/n), where rad() is A007947. - Enrique Pérez Herrero, Nov 13 2009
A175046(n) = a(n)*A073311(n). - Reinhard Zumkeller, Apr 05 2010
a(n) = floor(A000005(n^2)/A007425(n)). - Enrique Pérez Herrero, Apr 15 2010
a(A005117(n)) = 1; a(A013929(n)) = 0; a(n) = A013928(n + 1) - A013928(n). - Reinhard Zumkeller, Jul 05 2010
a(n) * A112526(n) = A063524(n). - Reinhard Zumkeller, Sep 16 2011
a(n) = mu(n) * lambda(n) = A008836(n) * A008683(n). - Enrique Pérez Herrero, Nov 29 2013
a(n) = Sum_{d|n} 2^omega(d)*mu(n/d). - Geoffrey Critzer, Feb 22 2015
a(n) = A085357(A156552(n)). - Antti Karttunen, Mar 06 2017
Limit_{n->oo} (1/n)*Sum_{j=1..n} a(j) = 6/Pi^2. - Andres Cicuttin, Aug 13 2017
a(1) = 1; a(n) = -Sum_{d|n, d < n} (-1)^bigomega(n/d) * a(d). - Ilya Gutkovskiy, Mar 10 2021

Deleted an unclear comment. - N. J. A. Sloane, May 30 2021

A126804 a(n) = (2n)! / (n-1)!.

Original entry on oeis.org

2, 24, 360, 6720, 151200, 3991680, 121080960, 4151347200, 158789030400, 6704425728000, 309744468633600, 15543540607795200, 841941782922240000, 48962152914554880000, 3042648073975910400000, 201220459292273541120000, 14110584707870682071040000

Offset: 1

Jonathan R. Love (japanada11(AT)yahoo.ca), Feb 22 2007

Old name was "Multiplying n X n integers above n".
a(n) = 2*A001814(n). - Zerinvary Lajos, May 03 2007
A179214(n) <= a(n). - Reinhard Zumkeller, Jul 05 2010
Product of the numbers from n to 2n. - Wesley Ivan Hurt, Dec 14 2015

			a(5) = 151200 because five digits above 5: (6, 7, 8, 9, 10), multiplied by five equals 5*(6*7*8*9*10) = 151200.

Robert Israel, Table of n, a(n) for n = 1..360
Elliot J. Carr and Matthew J. Simpson, Accurate and efficient calculation of finite response times for groundwater flow, arXiv:1707.06331 [physics.flu-dyn], 2017, p. 11.

Cf. A045943, A073838. - Reinhard Zumkeller, Jul 05 2010

Cf. A001814, A179214.

[Factorial(2*n)/Factorial(n-1) : n in [1..20]]; // Wesley Ivan Hurt, Dec 14 2015

a:=n->sum((count(Permutation(2*n+2),size=n+1)),j=0..n): seq(a(n), n=0..15); # Zerinvary Lajos, May 03 2007
seq(mul((n+k), k=0..n), n=1..16); # Zerinvary Lajos, Sep 21 2007
with(combstruct):with(combinat) :bin := {B=Union(Z,Prod(B,B))}: seq (count([B,bin,labeled],size=n)*(n-1), n=2..17); # Zerinvary Lajos, Dec 05 2007

Table[Pochhammer[n, n + 1], {n, 17}] (* Arkadiusz Wesolowski, Aug 13 2012 *)
Table[(2 n)!/(n - 1)!, {n, 20}] (* Wesley Ivan Hurt, Dec 14 2015 *)

a(n) = prod(k=n, 2*n, k); \\ Michel Marcus, Dec 15 2015

x='x+O('x^99); Vec(serlaplace(2*x/(1-4*x)^(3/2))) \\ Altug Alkan, Mar 11 2018

a(n) = (2n)! / (n-1)!.
a(n) = Product_{i=n..2n} i. - Wesley Ivan Hurt, Dec 14 2015
From Robert Israel, Dec 15 2015: (Start)
a(n) = (2*n*(2*n-1)/(n-1))*a(n-1).
E.g.f.: 2*x/(1-4*x)^(3/2). (End)
a(n) = Pochhammer(n,n+1). - Pedro Caceres, Mar 10 2018

New name from Wesley Ivan Hurt, Dec 15 2015

A179211 Number of squarefree numbers between n and 2*n (inclusive).

Original entry on oeis.org

2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 8, 9, 8, 9, 9, 11, 11, 13, 13, 15, 15, 15, 15, 15, 16, 16, 17, 19, 19, 20, 19, 21, 21, 22, 22, 24, 23, 24, 24, 25, 25, 26, 26, 27, 28, 29, 29, 30, 30, 32, 32, 34, 34, 36, 36, 38, 38, 38, 38, 39, 39, 38, 39, 41, 41, 42, 41, 43, 43, 44, 44, 46, 45, 45

Offset: 1

Reinhard Zumkeller, Jul 05 2010

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A005117, A008966, A013928, A035250, A061398, A179212, A179213, A179214.

a := n -> nops(select(issqrfree, [$n..(2*n)])):
seq(a(n), n=1..75); # Peter Luschny, Mar 02 2017

a[n_] := Select[Range[n, 2n], SquareFreeQ] // Length;
Array[a, 75] (* Jean-François Alcover, Jun 22 2018 *)

f(n)=my(s); forfactored(k=1,sqrtint(n), s += n\k[1]^2*moebius(k)); s
a(n)=f(2*n)-f(n-1) \\ Charles R Greathouse IV, Nov 05 2017

a(n) = Sum_{k=n..2*n} A008966(k).
a(n) > A035250(n) for n>2;
A179212(n) = a(n+1) - a(n);
a(n) = A013928(2*n+1) - A013928(n).
a(n) ~ (6/Pi^2) * n. - Amiram Eldar, Mar 03 2021

A179215 Product of squarefree numbers less than n+1.

Original entry on oeis.org

1, 1, 2, 6, 6, 30, 180, 1260, 1260, 1260, 12600, 138600, 138600, 1801800, 25225200, 378378000, 378378000, 6432426000, 6432426000, 122216094000, 122216094000, 2566537974000, 56463835428000, 1298668214844000, 1298668214844000, 1298668214844000, 33765373585944000

Offset: 0

Reinhard Zumkeller, Jul 05 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 0..680

Cf. A013928, A066779, A005117, A008966, A179214.

Cf. A000142, A000720, A001221, A025487, A034386.

a:= proc(n) option remember; `if`(n=0, 1,
      a(n-1)*`if`(issqrfree(n), n, 1))
    end:
seq(a(n), n=0..27);  # Alois P. Heinz, Sep 20 2021

With[{sfnos=Select[Range[50],SquareFreeQ]},Table[Times@@Select[sfnos, #Harvey P. Dale, Jun 13 2011 *)

a(n) = prod(k=1, n, if (issquarefree(k), k, 1)); \\ Michel Marcus, Sep 20 2021

a(n) = my(p=1); forsquarefree(x=1, n, p*=x[1]); p; \\ Michel Marcus, Sep 20 2021

a(n) = Product_{k=1..n} k^A008966(k).
A001221(a(n)) = A000720(n).
Subsequence of A025487.
A034386(n) <= a(n) <= A000142(n).
A179214(n) = a(2*n)/a(n-1) for n>0.

Definition corrected by Harvey P. Dale, Jun 13 2011

A073838 Product of primes p satisfying n <= p <= 2n.

Original entry on oeis.org

2, 6, 15, 35, 35, 77, 1001, 143, 2431, 46189, 46189, 96577, 96577, 7429, 215441, 6678671, 6678671, 392863, 14535931, 765049, 31367009, 1348781387, 1348781387, 2756205443, 2756205443, 2756205443, 146078888479, 146078888479, 146078888479, 297194980009

Offset: 1

Amarnath Murthy and Benoit Cloitre, Aug 12 2002

nonn

a(n) = A034386(2*n)/A034386(n-1); A179214(n) <= a(n). - Reinhard Zumkeller, Jul 05 2010

			a(7) = 1001 = 7*11*13 (product of primes between 7 and 14).

T. D. Noe, Table of n, a(n) for n = 1..500

Cf. A073837.

for n from 1 to 50 do l := 1:for j from n to 2*n do if isprime(j) then l := l*j:fi:od:a[n] := l:od:seq(a[j],j=1..50);

Table[Times @@ Select[Range[n, 2 n], PrimeQ], {n, 28}] (* Jayanta Basu, Aug 12 2013 *)

PARI
```
a(n)=prod(i=n,2*n,i^isprime(i))
```

More terms from Sascha Kurz, Aug 14 2002

Missing a(29) inserted by Andrew Howroyd, Feb 23 2018

A179213 Sum of squarefree numbers between n and 2*n (inclusive).

Original entry on oeis.org

3, 5, 14, 18, 28, 34, 55, 63, 80, 99, 132, 144, 170, 157, 202, 218, 285, 303, 378, 398, 481, 503, 527, 551, 551, 602, 629, 684, 799, 829, 922, 891, 1022, 1056, 1161, 1197, 1344, 1307, 1424, 1464, 1546, 1588, 1717, 1761, 1850, 1941, 2082, 2130, 2227, 2227

Offset: 1

Reinhard Zumkeller, Jul 05 2010

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005117, A045943, A066779, A073837, A179211, A179214.

Table[Total[Select[Range[n,2n],SquareFreeQ]],{n,50}] (* Harvey P. Dale, Apr 29 2017 *)

a(n) = Sum_{k=n..2*n} k*A008966(k).
A073837(n) <= a(n) <= A045943(n);
a(n) = A066779(2*n) - A066779(n-1).
a(n) ~ (9/Pi^2) * n^2 + O(n^(3/2)). - Amiram Eldar, Mar 07 2021

cp's OEIS Frontend

A008966 a(n) = 1 if n is squarefree, otherwise 0.

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A126804 a(n) = (2n)! / (n-1)!.

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A179211 Number of squarefree numbers between n and 2*n (inclusive).

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A179215 Product of squarefree numbers less than n+1.

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PARI

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A073838 Product of primes p satisfying n <= p <= 2n.

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Maple

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PARI

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A179213 Sum of squarefree numbers between n and 2*n (inclusive).

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Mathematica