A179211 -id:A179211 - 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

A013928 Number of (positive) squarefree numbers < n.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 6, 6, 6, 7, 8, 8, 9, 10, 11, 11, 12, 12, 13, 13, 14, 15, 16, 16, 16, 17, 17, 17, 18, 19, 20, 20, 21, 22, 23, 23, 24, 25, 26, 26, 27, 28, 29, 29, 29, 30, 31, 31, 31, 31, 32, 32, 33, 33, 34, 34, 35, 36, 37, 37, 38, 39, 39, 39, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 47, 47, 48, 49, 50, 50, 50, 51

Offset: 1

Henri Lifchitz

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

			a(10) = 6 because there are 6 squarefree numbers up to 10: 1, 2, 3, 5, 6, 7.
a(11) = 7 because there are 7 squarefree numbers up to 11: the numbers listed above for 10, plus 10 itself.
a(13) = 8 because the 12 X 12 matrix described in the first comment by Sharon Sela has rank 8. Rows 2,4,8 (the powers of two) are identical, rows 3,9 (the powers of three) are identical, and rows 6 and 12 (same prime factors) are identical. - _Geoffrey Critzer_, Dec 07 2014
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 0, 1, 0, 1, 0, 1, 0, 1, 0  1, 0, ...
1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, ...
1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...
1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, ...
1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, ...
1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, ...
1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...
1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, ...
1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, ...
1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, ...
.                                   .
.                                    .
.                                     .

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth edition (1979), Clarendon Press, pp. 269-270.
E. Landau, Über den Zusammenhang einiger neuer Sätze der analytischen Zahlentheorie, Wiener Sitzungberichte, Math. Klasse 115 (1906), pp. 589-632. Cited in Sándor, Mitrinović, & Crstici.
József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I. Springer, 2005. Section VI.18.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
Henri Cohen, Francois Dress, and Mahomed El Marraki, Explicit estimates for summatory functions linked to the Möbius μ-function, Funct. Approx. Comment. Math. 37:1 (2007), pp. 51-63.
G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number n, Q. J. Math., 48 (1917), pp. 76-92.
L. Moser and R. A. MacLeod, The error term for the squarefree integers, Canad. Math. Bull. vol. 9, no. 3, (1966).
K. Rogers, The Schnirelmann density of the squarefree integers, Proc. Amer. Math. Soc. 15 (1964), pp. 515-516.
A. M. Vaidya, On the order of the error function of the square free numbers, Proc. Nat. Inst. Sci. India Part A 32 (1966), pp. 196-201.
Eric Weisstein's World of Mathematics, Squarefree.

One less than A107079.
Cf. A005117, A002321, A057627, A179211, A000720, A081239, A066779, A179215, A284584.
Cf. A158819 Number of squarefree numbers <= n minus round(n/zeta(2)).

a013928 n = a013928_list !! (n-1)
a013928_list = scanl (+) 0 $ map a008966 [1..]
-- Reinhard Zumkeller, Aug 03 2012

ListTools:-PartialSums([0,seq(numtheory:-mobius(i)^2,i=1..100)]); # Robert Israel, Dec 11 2014

Accumulate[Table[Abs[MoebiusMu[n]], {n, 0, 79}]] (* Alonso del Arte, Oct 07 2012 *)
Accumulate[Table[If[SquareFreeQ[n],1,0],{n,0,80}]] (* Harvey P. Dale, Mar 06 2019 *)

a(n)=sum(i=1,n-1,if(issquarefree(i),1,0)) \\ Lifchitz

a(n)=n--;sum(k=1,sqrtint(n),moebius(k)*(n\k^2)) \\ Benoit Cloitre, Oct 25 2009

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

a(n)=n--; my(s); forsquarefree(k=1, sqrtint(n), s += n\k[1]^2*moebius(k)); s \\ Charles R Greathouse IV, Jan 08 2018

from sympy.ntheory.factor_  import core
def a(n): return sum ([1 for i in range(1, n) if core(i) == i]) # Indranil Ghosh, Apr 16 2017

from math import isqrt
from sympy import mobius
def A013928(n): return sum(mobius(k)*((n-1)//k**2) for k in range(1,isqrt(n-1)+1)) # Chai Wah Wu, Jan 03 2024

a(n) = Sum_{k = 1..n-1} mu(k)^2. - Vladeta Jovovic, May 18 2001
a(n) = Sum_{d = 1..floor(sqrt(n - 1))} mu(d)*floor((n - 1)/d^2) where mu(d) is the Moebius function (A008683). - Vladeta Jovovic, Apr 06 2001
Asymptotic formula (with error term): a(n) = Sum_{k = 1..n-1} mu(k)^2 = Sum_{k = 1..n-1} |mu(k)| = 6*n/Pi^2 + O(sqrt(n)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 20 2002
a(n) = Sum_{k = 0..n} if(k <= n-1, mu(n - k) mod 2, else 0; a(n + 1) = Sum_{k = 0..n} mu(n - k + 1) mod 2. - Paul Barry, May 10 2005
a(n + 1) = Sum_{k = 0..n} abs(mu(n - k + 1)). - Paul Barry, Jul 20 2005
a(n) = Sum_{k = 1..floor(sqrt(n))} mu(k)*floor(n/k^2). - Benoit Cloitre, Oct 25 2009
Landau proved that a(n) = 6*n/Pi^2 + o(sqrt(n)). - Charles R Greathouse IV, Feb 02 2016
Vaidya proved that a(n) = 6*n/Pi^2 + O(n^k) for any k > 2/5 on the Riemann hypothesis. - Charles R Greathouse IV, Feb 02 2016
a(n) = A107079(n)-1. - Antti Karttunen, Oct 07 2016
G.f.: Sum_{k>=1} mu(k)^2*x^(k+1)/(1 - x). - Ilya Gutkovskiy, Feb 06 2017
a(n+1) = n - A057627(n) - Antti Karttunen, Apr 17 2017

A061398 Number of squarefree integers between prime(n) and prime(n+1).

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 0, 2, 1, 1, 3, 2, 1, 1, 1, 3, 0, 3, 2, 0, 3, 1, 3, 4, 0, 1, 2, 0, 2, 6, 2, 2, 1, 5, 0, 2, 3, 2, 1, 3, 0, 6, 0, 2, 0, 7, 8, 1, 0, 2, 3, 0, 3, 3, 3, 3, 0, 2, 1, 1, 5, 7, 2, 0, 1, 9, 2, 4, 0, 0, 4, 3, 2, 2, 2, 2, 5, 2, 4, 6, 0, 5, 0, 4, 1, 3, 4, 1, 1, 2, 6, 4, 1, 4, 2, 2, 7, 0, 8, 4, 4, 3, 2, 1, 2

Offset: 1

Labos Elemer, Jun 07 2001

nonn

			Between 113 and 127 the 6 squarefree numbers are 114, 115, 118, 119, 122, 123, so a(30)=6.
From _Gus Wiseman_, Nov 06 2024: (Start)
The a(n) squarefree numbers for n = 1..16:
  1   2   3   4   5   6   7   8   9   10  11  12  13  14  15  16
  ---------------------------------------------------------------
  .   .   6   10  .   14  .   21  26  30  33  38  42  46  51  55
                      15      22          34  39              57
                                          35                  58
(End)

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A005117, A013928.
Cf. A179211. [Reinhard Zumkeller, Jul 05 2010]
Counting all composite numbers (not just squarefree) gives A046933.
The version for nonsquarefree numbers is A061399.
Zeros are A068360.
The version for prime-powers is A080101.
Partial sums are A337030.
The version for non-prime-powers is A368748.
Excluding prime(n+1) from the range gives A373198.
Ones are A377430.
Positives are A377431.
The version for perfect-powers is A377432.
The version for non-perfect-powers is A377433 + 2.
For squarefree numbers (A005117) between primes:
- length is A061398 (this sequence)
- min is A112926
- max is A112925
- sum is A373197
For squarefree numbers between powers of two:
- length is A077643 (except initial terms), partial sums A143658
- min is A372683, difference A373125, indices A372540, firsts of A372475
- max is A372889, difference A373126
- sum is A373123
For primes between powers of two:
- length is A036378
- min is A104080 or A014210, indices A372684 (firsts of A035100)
- max is A014234, difference A013603
- sum is A293697 (except initial terms)
Cf. A001223, A049093, A049094, A076259, A077641, A377434, A377436.

p:= 2:
for n from 1 to 200 do
  q:= nextprime(p);
A[n]:= nops(select(numtheory:-issqrfree, [$p+1..q-1]));
p:= q;
od:
seq(A[i],i=1..200); # Robert Israel, Jan 06 2017

a[n_] := Count[Range[Prime[n]+1, Prime[n+1]-1], _?SquareFreeQ];
Array[a, 100] (* Jean-François Alcover, Feb 28 2019 *)
Count[Range[#[[1]]+1,#[[2]]-1],?(SquareFreeQ[#]&)]&/@Partition[ Prime[ Range[120]],2,1] (* _Harvey P. Dale, Oct 14 2021 *)

{ n=0; q=2; forprime (p=3, prime(1001), a=0; for (i=q+1, p-1, a+=issquarefree(i)); write("b061398.txt", n++, " ", a); q=p ) } \\ Harry J. Smith, Jul 22 2009

a(n) = my(pp=prime(n)+1); sum(k=pp, nextprime(pp)-1, issquarefree(k)); \\ Michel Marcus, Feb 28 2019

from math import isqrt
from sympy import mobius, prime, nextprime
def A061398(n):
    p = prime(n)
    q = nextprime(p)
    r = isqrt(p-1)+1
    return sum(mobius(k)*((q-1)//k**2) for k in range(r,isqrt(q-1)+1))+sum(mobius(k)*((q-1)//k**2-(p-1)//k**2) for k in range(1,r))-1 # Chai Wah Wu, Jun 01 2024

a(n) = A013928(A000040(n+1)) - A013928(A000040(n)) - 1. - Robert Israel, Jan 06 2017

a(n) = A373198(n) - 1. - Gus Wiseman, Nov 06 2024

A061399 Number of nonsquarefree integers between primes prime(n) and prime(n+1).

Original entry on oeis.org

0, 1, 0, 2, 1, 1, 1, 1, 4, 0, 2, 1, 0, 2, 4, 2, 1, 2, 1, 1, 2, 2, 2, 3, 3, 0, 1, 1, 1, 7, 1, 3, 0, 4, 1, 3, 2, 1, 4, 2, 1, 3, 1, 1, 1, 4, 3, 2, 1, 1, 2, 1, 6, 2, 2, 2, 1, 3, 2, 0, 4, 6, 1, 1, 2, 4, 3, 5, 1, 3, 1, 4, 3, 3, 1, 3, 2, 1, 3, 3, 1, 4, 1, 1, 2, 2, 3, 2, 0, 1, 5, 3, 2, 3, 1, 3, 4, 1, 9, 1, 5, 2, 3, 0, 3

Offset: 1

Labos Elemer, Jun 07 2001

nonn

			Between 113 and 127 the 7 numbers which are not squarefree are {116,117,120,121,124,125,126}, so a(30)=7.
From _Gus Wiseman_, Dec 07 2024: (Start)
The a(n) nonsquarefree numbers for n = 1..15:
   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ----------------------------------------------------------
   .   4   .   8  12  16  18  20  24   .  32  40   .  44  48
               9                  25      36          45  49
                                  27                      50
                                  28                      52
(End)

Harry J. Smith, Table of n, a(n) for n=1..1000

Zeros are A068361.
First differences of A378086, restriction of A057627 to the primes.
Other classes (instead of nonsquarefree):
- For composite we have A046933, first differences of A065890.
- For squarefree see A061398, A068360, A071403, A373197, A373198, A377431.
- For prime power we have A080101.
- For non prime power we have A368748, see A378616.
- For perfect power we have A377432, zeros A377436.
- For non perfect power we have A377433, A029707.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A120327 gives the least nonsquarefree number >= n.
Cf. A007674, A053806, A072284, A073247, A179211, A224363.
Cf. A013928, A067535, A070321, A112925, A112926, A240473, A240474.
Cf. A045882, A377049, A377783, A378032, A378618.

Count[Range[#[[1]]+1,#[[2]]-1],?(!SquareFreeQ[#]&)]&/@Partition[Prime[Range[120]],2,1] (* _Harvey P. Dale, Mar 31 2024 *)

{ n=0; q=2; forprime (p=3, prime(1001), a=0; for (i=q+1, p-1, a+=!issquarefree(i)); write("b061399.txt", n++, " ", a); q=p ) } \\ Harry J. Smith, Jul 22 2009

a(n) = my(p=prime(n)); sum(k=p, nextprime(p+1), ! issquarefree(k)); \\ Michel Marcus, Dec 09 2024

from sympy import mobius, prime
def A061399(n): return sum(not mobius(m) for m in range(prime(n)+1,prime(n+1))) # Chai Wah Wu, Jul 20 2024

A035250 Number of primes between n and 2n (inclusive).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 3, 2, 3, 4, 4, 4, 4, 3, 4, 5, 5, 4, 5, 4, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 7, 7, 8, 8, 9, 10, 9, 9, 10, 10, 10, 10, 9, 10, 10, 10, 9, 10, 10, 11, 12, 12, 12, 13, 13, 14, 14, 14, 13, 13, 12, 12, 13, 13, 14, 14, 13, 14, 15, 15, 14, 14, 13, 14, 15

Offset: 1

Erich Friedman

nonn

By Bertrand's Postulate (proved by Chebyshev), there is always a prime between n and 2n, i.e., a(n) is positive for all n.
The number of primes in the interval [n,2*n) is the same sequence as this, except that a(1) = 0. - N. J. A. Sloane, Oct 18 2024
The smallest and largest primes between n and 2n inclusive are A007918 and A060308 respectively. - Lekraj Beedassy, Jan 01 2007
The number of partitions of 2n into exactly two parts with first part prime, n > 1. - Wesley Ivan Hurt, Jun 15 2013

			The primes between n = 13 and 2n = 26, inclusive, are 13, 17, 19, 23; so a(13) = 4.
a(5) = 2, since 2(5) = 10 has 5 partitions into exactly two parts: (9,1),(8,2),(7,3),(6,4),(5,5).  Two primes are among the first parts: 7 and 5.

Aigner, M. and Ziegler, G. Proofs from The Book (2nd edition). Springer-Verlag, 2001.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
International Mathematics Olympiad, Proof of Bertrand's Postulate [Via Wayback Machine]

Cf. A073837, A073838, A099802, A060715.
Related sequences:
Primes (p) and composites (c): A000040, A002808, A000720, A065855.
Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

a035250 n = sum $ map a010051 [n..2*n] -- Reinhard Zumkeller, Jan 08 2012

[#PrimesInInterval(n, 2*n): n in [1..80]]; // Bruno Berselli, Sep 05 2012

with(numtheory): A035250:=n->pi(2*n)-pi(n-1): seq(A035250(n), n=1..100); # Wesley Ivan Hurt, Aug 09 2014

f[n_] := PrimePi[2n] - PrimePi[n - 1]; Array[f, 76] (* Robert G. Wilson v, Dec 23 2012 *)

a(n)=primepi(2*n)-primepi(n-1) \\ Charles R Greathouse IV, Jul 01 2013

a(n) = A000720(2*n) - A000720(n-1); a(n) <= A179211(n). - Reinhard Zumkeller, Jul 05 2010
a(A059316(n)) = n and a(m) <> n for m < A059316(n). - Reinhard Zumkeller, Jan 08 2012
a(n) = sum(A010051(k): k=n..2*n). [Reinhard Zumkeller, Jan 08 2012]
a(n) = pi(2n) - pi(n-1). [Wesley Ivan Hurt, Jun 15 2013]

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

A179214 Product of squarefree numbers between n and 2*n (inclusive).

Original entry on oeis.org

2, 6, 90, 210, 2100, 4620, 140140, 300300, 5105100, 96996900, 4481256780, 9369900540, 243617414040, 18739801080, 1164544781400, 2406725881560, 2700346439110320, 5559536786403600, 7816708721683461600

Offset: 1

Reinhard Zumkeller, Jul 05 2010

nonn

A073838(n) <= a(n) <= A126804(n);

a(n) = A179215(2*n)/A179215(n-1).

			a(10) = 10*11*13*14*15*17*19 = 96996900;
a(11) = 11*13*14*15*17*19*21*22 = 4481256780;
a(12) = 13*14*15*17*19*21*22*23 = 9369900540.

Cf. A179211, A179213, A005117.

a(n) = PROD(k^A008966(k): n <= k <= 2*n).

Example corrected by Reinhard Zumkeller, Jul 19 2010

A378618 Sum of nonsquarefree numbers between prime(n) and prime(n+1).

Original entry on oeis.org

0, 4, 0, 17, 12, 16, 18, 20, 104, 0, 68, 40, 0, 89, 199, 110, 60, 127, 68, 72, 151, 161, 172, 278, 297, 0, 104, 108, 112, 849, 128, 403, 0, 579, 150, 461, 322, 164, 680, 351, 180, 561, 192, 196, 198, 819, 648, 449, 228, 232, 470, 240, 1472, 508, 521, 532, 270

Offset: 1

Gus Wiseman, Dec 09 2024

nonn

			The nonsquarefree numbers between prime(24) = 89 and prime(25) = 97 are {90, 92, 96}, so a(24) = 278.

For prime instead of nonsquarefree we have A001043.
For composite instead of nonsquarefree we have A054265.
Zeros are A068361.
A000040 lists the primes, differences A001223, seconds A036263.
A070321 gives the greatest squarefree number up to n.
A071403 counts squarefree numbers up to prime(n), restriction of A013928.
A120327 gives the least nonsquarefree number >= n.
A378086 counts nonsquarefree numbers up to prime(n), restriction of A057627.
For squarefree numbers (A005117, differences A076259) between primes:
- length is A061398, zeros A068360
- min is A112926, differences A378037
- max is A112925, differences A378038
- sum is A373197
For nonsquarefree numbers (A013929, differences A078147) between primes:
- length is A061399
- min is A377783 (differences A377784), union A378040
- max is A378032 (differences A378034), restriction of A378033 (differences A378036)
- sum is A378618 (this)
Cf. A007674, A045882, A053806, A067535, A072284, A073247, A179211, A224363, A373198, A377049, A377431.

Table[Total[Select[Range[Prime[n],Prime[n+1]],!SquareFreeQ[#]&]],{n,100}]

cp's OEIS Frontend

A179212 First differences of A179211.

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

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A013928 Number of (positive) squarefree numbers < n.

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A061398 Number of squarefree integers between prime(n) and prime(n+1).

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A061399 Number of nonsquarefree integers between primes prime(n) and prime(n+1).

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A035250 Number of primes between n and 2n (inclusive).

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