A061398 -id:A061398 - cp's OEIS frontend

A053797 Lengths of successive gaps between squarefree numbers.

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

Offset: 1

N. J. A. Sloane, Apr 07 2000

From Gus Wiseman, Jun 11 2024: (Start)
Also the length of the n-th maximal run of nonsquarefree numbers. These runs begin:
       4
     8   9
      12
      16
      18
      20
    24  25
    27  28
      32
      36
      40
    44  45
  48  49  50
(End)

			The first gap is at 4 and has length 1; the next starts at 8 and has length 2 (since neither 8 nor 9 are squarefree).

Peter Kagey, Table of n, a(n) for n = 1..10000
M. Filaseta and O. Trifonov, On Gaps between Squarefree Numbers. In Analytic Number Theory, Vol 85, 1990, Birkhäuser, Basel, pp. 235-253.
E. Fogels, On the average values of arithmetic functions, Proc. Cambridge Philos. Soc. 1941, 37: 358-372.
L. Marmet, First occurrences of squarefree gaps...
L. Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012. - From _N. J. A. Sloane_, Jan 01 2013
K. F. Roth, On the gaps between squarefree numbers, J. London Math. Soc. 1951 (2) 26:263-268.
Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

Gaps between terms of A005117.
For squarefree runs we have A120992, antiruns A373127 (firsts A373128).
For composite runs we have A176246 (rest of A046933), antiruns A373403.
For prime runs we have A251092 (rest of A175632), antiruns A027833.
Position of first appearance of n is A373199(n).
For antiruns instead of runs we have A373409.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A007674, A020754, A045882, A053806, A061398, A061399.

SF:= select(numtheory:-issqrfree,[$1..1000]):
map(`-`,select(`>`,SF[2..-1]-SF[1..-2],1),1); # Robert Israel, Sep 22 2015

ReplaceAll[Differences[Select[Range@384, SquareFreeQ]] - 1, 0 -> Nothing] (* Michael De Vlieger, Sep 22 2015 *)

Offset set to 1 by Peter Kagey, Sep 29 2015

A112925 Largest squarefree integer < the n-th prime.

Original entry on oeis.org

1, 2, 3, 6, 10, 11, 15, 17, 22, 26, 30, 35, 39, 42, 46, 51, 58, 59, 66, 70, 71, 78, 82, 87, 95, 97, 102, 106, 107, 111, 123, 130, 134, 138, 146, 149, 155, 161, 166, 170, 178, 179, 190, 191, 195, 197, 210, 222, 226, 227, 231, 238, 239, 249, 255, 262, 267, 269, 274, 278

Offset: 1

Leroy Quet, Oct 06 2005

nonn

			6 is the largest squarefree less than the 4th prime, 7. So a(4) = 6.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

For prime powers instead of squarefree numbers we have A065514, opposite A345531.
Restriction of A070321 (differences A378085) to the primes; see A378619.
The opposite is A112926, differences A378037.
Subtracting each term from prime(n) gives A240473, opposite A240474.
For nonsquarefree numbers we have A378033, differences A378036, see A378034, A378032.
For perfect powers we have A378035.
First differences are A378038.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259.
A013928 counts squarefree numbers up to n - 1.
A013929 lists the nonsquarefree numbers, differences A078147.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
A112929 counts squarefree numbers up to prime(n).
Cf. A061400, A112928, A112930.
Cf. A007674, A007947, A045882, A053797, A053806, A067535, A072284, A073247, A081217, A175216, A280892, A377783.

with(numtheory): a:=proc(n) local p,B,j: p:=ithprime(n): B:={}: for j from 1 to p-1 do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: B[nops(B)] end: seq(a(m),m=1..75); # Emeric Deutsch, Oct 14 2005

With[{k = 120}, Table[SelectFirst[Range[Prime@ n - 1, Prime@ n - Min[Prime@ n - 1, k], -1], SquareFreeQ], {n, 60}]] (* Michael De Vlieger, Aug 16 2017 *)

a(n,p=prime(n))=while(!issquarefree(p--),); p \\ Charles R Greathouse IV, Aug 16 2017

a(n) = prime(n) - A240473(n). - Gus Wiseman, Jan 10 2025

More terms from Emeric Deutsch, Oct 14 2005

A112926 Smallest squarefree integer > the n-th prime.

Original entry on oeis.org

3, 5, 6, 10, 13, 14, 19, 21, 26, 30, 33, 38, 42, 46, 51, 55, 61, 62, 69, 73, 74, 82, 85, 91, 101, 102, 105, 109, 110, 114, 129, 133, 138, 141, 151, 154, 158, 165, 170, 174, 181, 182, 193, 194, 199, 201, 213, 226, 229, 230, 235, 241, 246, 253, 258, 265, 271, 273

Offset: 1

Leroy Quet, Oct 06 2005

nonn

			10 is the smallest squarefree number greater than the 4th prime, 7. So a(4) = 10.
From _Gus Wiseman_, Dec 07 2024: (Start)
The first number line below shows the squarefree numbers. The second shows the primes:
--1--2--3-----5--6--7-------10-11----13-14-15----17----19----21-22-23-------26--
=====2==3=====5=====7==========11====13==========17====19==========23===========
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A061400, A112928, A112929, A112930.
Restriction of A067535, differences A378087.
The unrestricted opposite is A070321, differences A378085.
The opposite is A112925, differences A378038.
Subtracting prime(n) from each term gives A240474, opposite A240473.
For nonsquarefree we have A377783, restriction of A120327.
The nonsquarefree differences are A377784, restriction of A378039.
First differences are A378037.
For perfect power we have A378249, A378617, A378250, A378251.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
Cf. A007674, A013928, A053797, A053806, A072284, A073247, A175216, A280892, A345531, A378032, A378618.

with(numtheory): a:=proc(n) local p,B,j: p:=ithprime(n): B:={}: for j from p+1 to p+20 do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: B[1] end: seq(a(m),m=1..75); # Emeric Deutsch, Oct 10 2005

Do[k = Prime[n] + 1; While[ !SquareFreeQ[k], k++ ]; Print[k], {n, 1, 100}] (* Ryan Propper, Oct 10 2005 *)
With[{k = 120}, Table[SelectFirst[Range[Prime@ n + 1, Prime@ n + k], SquareFreeQ], {n, 58}]] (* Michael De Vlieger, Aug 16 2017 *)

a(n,p=prime(n))=while(!issquarefree(p++),); p \\ Charles R Greathouse IV, Aug 16 2017

a(n) = prime(n) + A240474(n). - Gus Wiseman, Dec 07 2024

More terms from Ryan Propper and Emeric Deutsch, Oct 10 2005

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

A070321 Greatest squarefree number <= n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 7, 7, 10, 11, 11, 13, 14, 15, 15, 17, 17, 19, 19, 21, 22, 23, 23, 23, 26, 26, 26, 29, 30, 31, 31, 33, 34, 35, 35, 37, 38, 39, 39, 41, 42, 43, 43, 43, 46, 47, 47, 47, 47, 51, 51, 53, 53, 55, 55, 57, 58, 59, 59, 61, 62, 62, 62, 65, 66, 67, 67, 69, 70, 71, 71

Offset: 1

Benoit Cloitre, May 11 2002

a(n) = Max( core(k) : k=1,2,3,...,n ) where core(x) is the squarefree part of x (the smallest integer such that x*core(x) is a square).

			From _Gus Wiseman_, Dec 10 2024: (Start)
The squarefree numbers <= n are the following columns, with maxima a(n):
  1  2  3  3  5  6  7  7  7  10  11  11  13  14  15  15
     1  2  2  3  5  6  6  6  7   10  10  11  13  14  14
        1  1  2  3  5  5  5  6   7   7   10  11  13  13
              1  2  3  3  3  5   6   6   7   10  11  11
                 1  2  2  2  3   5   5   6   7   10  10
                    1  1  1  2   3   3   5   6   7   7
                             1   2   2   3   5   6   6
                                 1   1   2   3   5   5
                                         1   2   3   3
                                             1   2   2
                                                 1   1
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Mayank Pandey, Squarefree numbers in short intervals, arXiv preprint (2024). arXiv:2401.13981 [math.NT]

Cf. A007947, A076260.
Cf. A081217, A081218, A081210.
The distinct terms are A005117 (the squarefree numbers).
The opposite version is A067535, differences A378087.
The run-lengths are A076259.
Restriction to the primes is A112925; see A378038, A112926, A378037.
For nonsquarefree we have A378033; see A120327, A378036, A378032, A377783.
First differences are A378085.
Subtracting each term from n gives A378619.
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, A053806, A072284, A073247, A112929, A240473.

A070321 := proc(n)
    local a;
    for a from n by -1 do
        if issqrfree(a) then
            return a;
        end if;
    end do:
end proc:
seq(A070321(n),n=1..100) ; # R. J. Mathar, May 25 2023

a[n_] :=For[ k = n, True, k--, If[ SquareFreeQ[k], Return[k]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 27 2013 *)
gsfn[n_]:=Module[{k=n},While[!SquareFreeQ[k],k--];k]; Array[gsfn,80] (* Harvey P. Dale, Mar 27 2013 *)

a(n) = while (! issquarefree(n), n--); n; \\ Michel Marcus, Mar 18 2017

from itertools import count
from sympy import factorint
def A070321(n): return next(m for m in count(n,-1) if max(factorint(m).values(),default=0)<=1) # Chai Wah Wu, Dec 04 2024

a(n) = n - o(n^(1/5)) by a result of Pandey. - Charles R Greathouse IV, Dec 04 2024

a(n) = A005117(A013928(n+1)). - Ridouane Oudra, Jul 26 2025

New description from Reinhard Zumkeller, Oct 03 2002

A376593 Second differences of consecutive nonsquarefree numbers (A013929). First differences of A078147.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 01 2024

sign

The range is {-3, -2, -1, 0, 1, 2, 3}.

			The nonsquarefree numbers (A013929) are:
  4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, ...
with first differences (A078147):
  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, ...
with first differences (A376593):
  -3, 2, 1, -2, 0, 2, -3, 1, -1, 3, 0, 0, 0, -3, 2, -2, 0, 1, 0, 0, 2, -1, -2, ...

The version for A000002 is A376604, first differences of A054354.
The first differences were A078147.
Zeros are A376594, complement A376595.
A000040 lists the prime numbers, differences A001223.
A005117 lists squarefree numbers, differences A076259.
A064113 lists positions of adjacent equal prime gaps.
A114374 counts partitions into nonsquarefree numbers.
A246655 lists prime-powers exclusive, inclusive A000961.
A333254 lists run-lengths of differences between consecutive primes.
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
For nonsquarefree numbers: A013929 (terms), A078147 (first differences), A376594 (inflections and undulations), A376595 (nonzero curvature).
Cf. A061398, A053797, A053806, A120992, A182853, A251092, A373198, A375707, A376305, A376306, A376311, A376312, A376655.

Differences[Select[Range[100],!SquareFreeQ[#]&],2]

from math import isqrt
from sympy import mobius, factorint
def A376593(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)
    k = next(i for i in range(1,5) if any(d>1 for d in factorint(m+i).values()))
    return next(i for i in range(1-k,5-k) if any(d>1 for d in factorint(m+(k<<1)+i).values())) # Chai Wah Wu, Oct 02 2024

A120327 Smallest nonsquarefree number >= n.

Original entry on oeis.org

4, 4, 4, 4, 8, 8, 8, 8, 9, 12, 12, 12, 16, 16, 16, 16, 18, 18, 20, 20, 24, 24, 24, 24, 25, 27, 27, 28, 32, 32, 32, 32, 36, 36, 36, 36, 40, 40, 40, 40, 44, 44, 44, 44, 45, 48, 48, 48, 49, 50, 52, 52, 54, 54, 56, 56, 60, 60, 60, 60, 63, 63, 63, 64, 68, 68, 68, 68, 72, 72, 72, 72

Offset: 1

Zak Seidov, Aug 16 2006

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

For squarefree instead of nonsquarefree we have A067535, differences A378087.
The opposite for squarefree is A070321, differences A378085.
The run-lengths are A078147 if we prepend 4, differences A376593.
The restriction to primes is A377783 (union A378040), differences A377784.
The opposite is A378033 (differences A378036), for prime powers A031218.
First differences are A378039 if we assume that a(1) = 1.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
Restriction to primes: A112925, A112926, A337030, A378032, A378034, A378037, A378038, A378082, A378084.
Cf. A000015, A007675, A013928, A053797, A053806, A068781, A072284, A073247, A120992, A224363.

Table[NestWhile[ #+1&,n,SquareFreeQ],{n,100}] (* simplified by Harvey P. Dale, Apr 08 2014 *)

A373198 Number of squarefree numbers from prime(n) to prime(n+1) - 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 29 2024

nonn

			This is the sequence of row-lengths of A005117 treated as a triangle with row-sums A373197:
   2
   3
   5   6
   7  10
  11
  13  14  15
  17
  19  21  22
  23  26
  29  30
  31  33  34  35
  37  38  39
  41  42
  43  46
  47  51
  53  55  57  58

Counting all numbers (not just squarefree) gives A001223, sum A371201.
For composite instead of squarefree we have A046933.
For squarefree numbers (A005117) between primes:
- sum is A373197
- length is A373198 (this sequence) = A061398 - 1
- min is A000040
- max is A112925, opposite A112926
For squarefree numbers between powers of two:
- sum is A373123
- length is A077643, partial sums A143658
- min is A372683, delta A373125, indices A372540, firsts of A372475
- max is A372889, delta A373126
For primes between powers of two:
- sum is A293697 (except initial terms)
- length is A036378
- min is A104080 or A014210, indices A372684 (firsts of A035100)
- max is A014234, delta A013603
Cf. A372473 (firsts of A372472), A372541 (firsts of A372433).
Cf. A049093, A049094, A076259, A077641.

Table[Length[Select[Range[Prime[n],Prime[n+1]-1],SquareFreeQ]],{n,100}]

from math import isqrt
from sympy import prime, nextprime, mobius
def A373198(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)) # Chai Wah Wu, Jun 01 2024

a(n) = A061398(n) + 1.

A068360 Numbers n such that there is no squarefree number between prime(n) and prime(n+1).

Original entry on oeis.org

1, 2, 5, 7, 17, 20, 25, 28, 35, 41, 43, 45, 49, 52, 57, 64, 69, 70, 81, 83, 98, 109, 120, 140, 144, 152, 171, 173, 176, 178, 182, 190, 206, 215, 225, 229, 230, 236, 253, 256, 261, 262, 277, 286, 294, 296, 302, 307, 315, 318, 323, 336, 346, 348, 373, 377, 390

Offset: 1

Benoit Cloitre, Feb 28 2002

nonn

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

Cf. A061398.

Join[{1},PrimePi/@Transpose[With[{prs=Partition[Prime[Range[ 400]],2,1]}, Select[ prs,Union[SquareFreeQ/@Range[#[[1]]+1,#[[2]]-1]]=={False}&]]][[1]]] (* Harvey P. Dale, Jan 24 2013 *)

n such that A061398(n)=0

A077643 Number of squarefree integers in closed interval [2^n, -1 + 2*2^n], i.e., among 2^n consecutive numbers beginning with 2^n.

Original entry on oeis.org

1, 2, 3, 5, 9, 19, 39, 79, 157, 310, 621, 1246, 2491, 4980, 9958, 19924, 39844, 79672, 159365, 318736, 637457, 1274916, 2549816, 5099651, 10199363, 20398663, 40797299, 81594571, 163189087, 326378438, 652756861, 1305513511, 2611026987, 5222053970, 10444108084

Offset: 0

Labos Elemer, Nov 14 2002

nonn

Number of squarefree numbers with binary expansion of length n, or with n bits. The sum of these numbers is given by A373123. - Gus Wiseman, Jun 02 2024

			For n=4: among the 16 numbers of {16, ..., 31}, nine are squarefree [17, 19, 21, 22, 23, 26, 29, 30, 31], so a(4) = 9.

Amiram Eldar, Table of n, a(n) for n = 0..63 (calculated from the b-file at A143658)

Cf. A077641, A077642.
Partial sums (except first term) are A143658.
Run-lengths of A372475.
The minimum is A372683, delta A373125, indices A372540.
The maximum is A372889 (except at n=1), delta A373126, indices A143658.
Row-sums are A373123.
A005117 lists squarefree numbers, first differences A076259.
A053797 gives nonempty lengths of exclusive gaps between squarefree numbers.
A029837 counts bits, row-lengths of A030190 and A030308.
For primes between powers of 2:
- sum A293697
- length A036378 or A162145
- min A104080 or A014210, delta A092131, indices A372684
- max A014234, delta A013603, indices A007053
For squarefree numbers between primes:
- sum A373197
- length A373198 = A061398 - 1
- min A000040
- max A112925 (delta A240473), opposite A112926 (delta A240474)
Cf. A010036, A029931, A035100, A049093-A049096, A372473 (firsts of A372472), A372541 (firsts of A372433).

Table[Apply[Plus, Table[Abs[MoebiusMu[2^w+j]], {j, 0, 2^w-1}]], {w, 0, 15}]
(* second program *)
Length/@Split[IntegerLength[Select[Range[10000],SquareFreeQ],2]]//Most (* Gus Wiseman, Jun 02 2024 *)

{ a(n) = sum(m=1,sqrtint(2^(n+1)-1), moebius(m) * ((2^(n+1)-1)\m^2 - (2^n-1)\m^2) ) } \\ Max Alekseyev, Oct 18 2008

a(n) = Sum_{j=0..-1+2^n} abs(mu(2^n + j)).

a(n)/2^n approaches 1/zeta(2), so limiting sequence is floor(2^n/zeta(2)), n >= 0. - Wouter Meeussen, May 25 2003

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 12 2003
More terms from Wouter Meeussen, May 25 2003
a(25)-a(32) from Max Alekseyev, Oct 18 2008
a(33)-a(34) from Amiram Eldar, Jul 17 2024

cp's OEIS Frontend

A053797 Lengths of successive gaps between squarefree numbers.

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A112925 Largest squarefree integer < the n-th prime.

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A112926 Smallest squarefree integer > the n-th prime.

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

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A070321 Greatest squarefree number <= n.

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A376593 Second differences of consecutive nonsquarefree numbers (A013929). First differences of A078147.

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A120327 Smallest nonsquarefree number >= n.

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