A013928 -id:A013928 - cp's OEIS frontend

A378619 Distance between n and the greatest squarefree number <= n.

0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 2, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 2, 3, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 2, 3, 0, 0, 0, 1, 0

Offset: 1

Gus Wiseman, Dec 12 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..30000

Positions of 0 are A005117.
Positions of first appearances are A020755 - 1.
Positions of 1 are A053806.
Subtracting each term from n gives A070321.
The opposite version is A081221.
Restriction to the primes is A240473, opposite A240474.
A013929 lists the nonsquarefree numbers, differences A078147.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
Cf. A007674, A013928, A053797, A072284, A073247, A076259, A280892.
Cf. A007947, A067535, A081217, A112925.

Table[n-NestWhile[#-1&,n,!SquareFreeQ[#]&],{n,100}]

A378619(n) = forstep(k=n,1,-1,if(issquarefree(k), return(n-k))); \\ Antti Karttunen, Jan 29 2025

from itertools import count
from sympy import factorint
def A378619(n): return n-next(m for m in count(n,-1) if max(factorint(m).values(),default=0)<=1) # Chai Wah Wu, Dec 14 2024

a(n) = n - A070321(n).

Data section extended to a(105) by Antti Karttunen, Jan 29 2025

A072358 Number of cubefree numbers <= n which are not squarefree.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 11, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19

Offset: 1

Reinhard Zumkeller, Jul 18 2002

nonn

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

Cf. A060431, A013928.
Cf. A008966, A212793.
Cf. A067259, A004709, A005117, A059956, A088453.

a072358 n = a072358_list !! (n-1)
a072358_list = scanl1 (+) $
   zipWith (*) a212793_list $ map (1 -) a008966_list
-- Reinhard Zumkeller, May 27 2012

Accumulate @ Table[Boole[Max @ FactorInteger[n][[;; , 2]] == 2], {n, 1, 100}] (* Amiram Eldar, Feb 16 2021 *)

from math import isqrt
from sympy import mobius, integer_nthroot
def A072358(n): return sum(mobius(k)*(n//k**3-n//k**2) for k in range(1, integer_nthroot(n,3)[0]+1))-sum(mobius(k)*(n//k**2) for k in range(integer_nthroot(n,3)[0]+1,isqrt(n)+1)) # Chai Wah Wu, Aug 12 2024

a(n) = A060431(n) - A013928(n+1).
a(n) = Sum_{k=1..n} (A212793(k) * (1 - A008966(k))). - Reinhard Zumkeller, May 27 2012
a(n) ~ c * n, where c = 1/zeta(3) - 1/zeta(2) = A088453 - A059956 = 0.22398... - Amiram Eldar, Feb 16 2021

A072490 Number of squarefree numbers (excluding 1) less than n.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 4, 5, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 11, 12, 12, 13, 14, 15, 15, 15, 16, 16, 16, 17, 18, 19, 19, 20, 21, 22, 22, 23, 24, 25, 25, 26, 27, 28, 28, 28, 29, 30, 30, 30, 30, 31, 31, 32, 32, 33, 33, 34, 35, 36, 36, 37, 38, 38, 38, 39

Offset: 1

Amarnath Murthy, Jul 14 2002

nonn

			a(10) = 5 as the squarefree numbers less than 10 are 2,3,5,6 and 7.

Cf. A005117, A013928, A144338.

a(n) = sum(k=2, n-1, issquarefree(k)); \\ Michel Marcus, Sep 15 2019

from math import isqrt
from sympy import mobius
def A072490(n): return int(sum(mobius(k)*((n-1)//k**2) for k in range(1, isqrt(n-1)+1)))-1 if n>1 else 0 # Chai Wah Wu, Aug 19 2024

a(n) = A013928(n) - 1, n > 1.

G.f.: (x/(1 - x)) * Sum_{k>=2} mu(k)^2*x^k. - Ilya Gutkovskiy, Sep 14 2019

Name clarified and more terms from Ilya Gutkovskiy, Sep 14 2019

A072778 Number of powers of squarefree numbers <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 11, 12, 13, 14, 15, 16, 16, 17, 17, 18, 19, 20, 20, 21, 22, 23, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 34, 35, 36, 37, 37, 37, 38, 39, 39, 40, 40, 41, 41, 42, 42, 43, 43, 44, 45, 46, 46, 47, 48, 48, 49, 50, 51, 52, 52, 53, 54, 55, 55

Offset: 1

Reinhard Zumkeller, Jul 10 2002

nonn

Cf. A072774, A013928.

from math import isqrt
from sympy import mobius, integer_nthroot
def A072778(n):
    def f(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    return 2-(m:=n.bit_length())+sum(f(integer_nthroot(n,k)[0]) for k in range(1,m)) # Chai Wah Wu, Aug 19 2024

A072980 Numerator of b(n) = Sum_{k'<=n} 1/k', where k' denotes the squarefree numbers.

Original entry on oeis.org

1, 3, 11, 11, 61, 11, 82, 82, 82, 171, 1951, 1951, 26133, 13424, 41273, 41273, 716656, 716656, 13871719, 13871719, 4700888, 9548741, 222854273, 222854273, 222854273, 112857219, 112857219, 112857219, 3310041496, 20075905417

Offset: 1

Benoit Cloitre, Aug 21 2002

b(n) was used by Niven in the 70's for an alternative proof of the divergence of the sum of reciprocal of primes.

			Fractions begin with 1, 3/2, 11/6, 11/6, 61/30, 11/5, 82/35, 82/35, 82/35, 171/70, 1951/770, 1951/770, ...

Ivan Niven, A Proof of the Divergence of sum 1/p, The American Mathematical Monthly, Vol. 78, No. 3 (Mar., 1971), pp. 272-273.

Cf. A005117, A013928, A072983 (denominators), A354417.

Accumulate[Table[If[SquareFreeQ[n], 1, 0]/n, {n, 1, 50}]] // Numerator (* Amiram Eldar, Apr 22 2025 *)

a(n) = numerator(sum(k=1, n, issquarefree(k)/k)); \\ Michel Marcus, Nov 28 2013

a(n) = A354417(A013928(n+1)). - Amiram Eldar, Apr 22 2025

A072983 Denominator of b(n) = Sum_{k'<=n} 1/k', where k' denotes the squarefree numbers.

Original entry on oeis.org

1, 2, 6, 6, 30, 5, 35, 35, 35, 70, 770, 770, 10010, 5005, 15015, 15015, 255255, 255255, 4849845, 4849845, 1616615, 3233230, 74364290, 74364290, 74364290, 37182145, 37182145, 37182145, 1078282205, 6469693230, 200560490130, 200560490130

Offset: 1

Benoit Cloitre, Aug 21 2002

b(n) was used by Niven in the 1970's for an alternative proof of the divergence of the sum of reciprocals of the primes.

Ivan Niven, A Proof of the Divergence of sum 1/p, The American Mathematical Monthly, Vol. 78, No. 3 (Mar., 1971), pp. 272-273.

Cf. A005117, A013928, A072980 (numerators), A354418.

Accumulate[Table[If[SquareFreeQ[n], 1, 0]/n, {n, 1, 50}]] // Denominator (* Amiram Eldar, Apr 22 2025 *)

a(n)=denominator(sum(k=1,n,issquarefree(k)/k))

a(n) = A354418(A013928(n+1)). - Amiram Eldar, Apr 22 2025

A160764 a(n) = n-th squarefree number minus round(n*zeta(2)).

Original entry on oeis.org

-1, -1, -2, -2, -2, -3, -2, -2, -2, -2, -3, -3, -2, -2, -3, -3, -2, -1, -1, -2, -2, -2, -3, -2, -3, -4, -3, -4, -5, -3, -4, -2, -1, -1, -1, -1, -2, -2, -2, -1, -1, -2, -2, -2, -3, -3, -3, -2, -3, -3, -2, -3, -2, -3, -3, -3, -3, -2, -3, -4, -3, -1, -2, -2, -2, -3, -3, -3, -4

Offset: 1

Daniel Forgues, May 26 2009

sign

Race between the n-th squarefree number and round(n*zeta(2)).

Daniel Forgues, Table of n, a(n) for n=1..60794

Cf. A005117 Squarefree numbers.
Cf. A013929 Nonsquarefree numbers.
Cf. A013928 Number of squarefree numbers < n.
Cf. A158819 Number of squarefree numbers <= n minus round(n/zeta(2)).

Since zeta(2) = Sum_{i>=1}, 1/(i^2) = (Pi^2)/6, we get:
a(n) = A005117(n) - n * Sum_{i>=1}, 1/(i^2) = O(sqrt(n));
a(n) = A005117(n) - n * (Pi^2)/6 = O(sqrt(n)).

A238646 Number of primes p < n such that the number of squarefree numbers among 1, ..., n-p is prime.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 2, 2, 3, 1, 2, 1, 3, 1, 3, 1, 4, 2, 3, 2, 5, 4, 5, 1, 3, 3, 4, 2, 5, 3, 4, 5, 8, 3, 5, 1, 5, 5, 7, 3, 5, 2, 6, 3, 6, 6, 9, 4, 8, 7, 7, 6, 7, 4, 6, 7, 8, 5, 6, 4, 7, 8, 9, 6, 6, 6, 9, 5, 7, 4, 8, 6, 10, 6, 5, 8, 11, 7, 10, 6

Offset: 1

Zhi-Wei Sun, Mar 02 2014

nonn

Conjecture: a(n) > 0 for all n > 3, and a(n) = 1 only for n = 4, 10, 12, 14, 16, 24, 36.
This is analog of the conjecture in A237705 for squarefree numbers.
We have verified the conjecture for n up to 60000.

			a(10) = 1 since 7 and 3 are both prime, and there are exactly 3 squarefree numbers among 1, ..., 10-7.
a(36) = 1 since 17 and 13 are both prime, and there are exactly 13 squarefree numbers among 1, ..., 36-17 (namely, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Cf. A000040, A005117, A013928, A237705, A237768, A237769, A238645.

s[n_]:=Sum[If[SquareFreeQ[k],1,0],{k,1,n}]
a[n_]:=Sum[If[PrimeQ[s[n-Prime[k]]],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,80}]

A245268 Sum of binomial(n,k) over squarefree k.

Original entry on oeis.org

1, 3, 7, 14, 26, 48, 92, 184, 375, 758, 1497, 2884, 5461, 10286, 19507, 37584, 73866, 147987, 301075, 618794, 1278116, 2640993, 5439593, 11138764, 22640100, 45644797, 91293390, 181301470, 358024924, 704359427, 1383415456, 2718141072, 5351701032, 10570658330

Offset: 1

Eric M. Schmidt, Jul 15 2014

nonn

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
J. E. Nymann and W. J. Leahey, On the probability that an integer chosen according to the binomial distribution be k-free, Rocky Mountain Journal of Mathematics 7 (1977), no. 4, 769-774.

Cf. A013928, A060431, A245269.

a[n_] := Sum[Binomial[n, k], {k, Select[Range[n], SquareFreeQ]}]; Array[a, 34] (* Amiram Eldar, May 25 2025 *)

a(n) = sum(k=1, n, if (issquarefree(k), binomial(n,k), 0)); \\ Michel Marcus, Jul 16 2014

def A235268(n) : return sum(binomial(n,k) for k in range(1,n+1) if is_squarefree(k))

a(n) ~ 2^n/zeta(2). [Take p = 1/2 in Nymann and Leahey.]

A245269 Sum of binomial(n,k) over cubefree k.

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 254, 502, 978, 1882, 3600, 6904, 13380, 26332, 52664, 106744, 218232, 447736, 917760, 1873312, 3799920, 7653136, 15306272, 30429856, 60234528, 118956831, 234885092, 464595690, 921868388, 1836393687, 3672648928, 7369572624, 14821243232

Offset: 1

Eric M. Schmidt, Jul 15 2014

nonn

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
J. E. Nymann and W. J. Leahey, On the probability that an integer chosen according to the binomial distribution be k-free, Rocky Mountain Journal of Mathematics 7 (1977), no. 4, 769-774.

Cf. A013928, A060431, A245268.

cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 3 &]; a[n_] := Sum[Binomial[n, k], {k, Select[Range[n], cubeFreeQ]}]; Array[a, 34] (* Amiram Eldar, May 25 2025 *)

def A245269(n) : return sum(binomial(n,k) for k in range(1,n+1) if all(m <= 2 for (p,m) in factor(k)))

a(n) ~ 2^n/zeta(3). [Take p = 1/2 in Nymann and Leahey.]

cp's OEIS Frontend

A378619 Distance between n and the greatest squarefree number <= n.

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A072358 Number of cubefree numbers <= n which are not squarefree.

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A072490 Number of squarefree numbers (excluding 1) less than n.

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A072778 Number of powers of squarefree numbers <= n.

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A072980 Numerator of b(n) = Sum_{k'<=n} 1/k', where k' denotes the squarefree numbers.

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A072983 Denominator of b(n) = Sum_{k'<=n} 1/k', where k' denotes the squarefree numbers.

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A160764 a(n) = n-th squarefree number minus round(n*zeta(2)).

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A238646 Number of primes p < n such that the number of squarefree numbers among 1, ..., n-p is prime.

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