A013928 -id:A013928 - cp's OEIS frontend

A071403 Which squarefree number is prime? a(n)-th squarefree number equals n-th prime.

2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 20, 24, 27, 29, 31, 33, 37, 38, 42, 45, 46, 50, 52, 56, 61, 62, 64, 67, 68, 71, 78, 81, 84, 86, 92, 93, 96, 100, 103, 105, 109, 110, 117, 118, 121, 122, 130, 139, 141, 142, 145, 149, 150, 154, 158, 162, 166, 167, 170, 172, 174, 180

Offset: 1

Labos Elemer, May 24 2002

nonn

Also the number of squarefree numbers <= prime(n). - Gus Wiseman, Dec 08 2024

			a(25)=61 because A005117(61) = prime(25) = 97.
From _Gus Wiseman_, Dec 08 2024: (Start)
The squarefree numbers up to prime(n) begin:
n = 1  2  3  4   5   6   7   8   9  10
    ----------------------------------
    2  3  5  7  11  13  17  19  23  29
    1  2  3  6  10  11  15  17  22  26
       1  2  5   7  10  14  15  21  23
          1  3   6   7  13  14  19  22
             2   5   6  11  13  17  21
             1   3   5  10  11  15  19
                 2   3   7  10  14  17
                 1   2   6   7  13  15
                     1   5   6  11  14
                         3   5  10  13
                         2   3   7  11
                         1   2   6  10
                             1   5   7
                                 3   6
                                 2   5
                                 1   3
                                     2
                                     1
The column-lengths are a(n).
(End)

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

Cf. A000290, A013928.
The strict version is A112929.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A070321 gives the greatest squarefree number up to n.
Other families: A014689, A027883, A378615, A065890.
Squarefree numbers between primes: A061398, A068360, A373197, A373198, A377430, A112925, A112926.
Nonsquarefree numbers: A057627, A378086, A061399, A068361, A120327, A377783, A378032, A378033.
Cf. A046933, A049093, A053797, A072284, A077641, A224363, A337030, A345531, A008864.

Position[Select[Range[300], SquareFreeQ], ?PrimeQ][[All, 1]] (* _Michael De Vlieger, Aug 17 2023 *)

lista(nn)=sqfs = select(n->issquarefree(n), vector(nn, i, i)); for (i = 1, #sqfs, if (isprime(sqfs[i]), print1(i, ", "));); \\ Michel Marcus, Sep 11 2013

a(n,p=prime(n))=sum(k=1, sqrtint(p), p\k^2*moebius(k)) \\ Charles R Greathouse IV, Sep 13 2013

a(n,p=prime(n))=my(s); forfactored(k=1, sqrtint(p), s+=p\k[1]^2*moebius(k)); s \\ Charles R Greathouse IV, Nov 27 2017

first(n)=my(v=vector(n),pr,k); forsquarefree(m=1,n*logint(n,2)+3, k++; if(m[2][,2]==[1]~, v[pr++]=k; if(pr==n, return(v)))) \\ Charles R Greathouse IV, Jan 08 2018

from math import isqrt
from sympy import prime, mobius
def A071403(n): return (p:=prime(n))+sum(mobius(k)*(p//k**2) for k in range(2,isqrt(p)+1)) # Chai Wah Wu, Jul 20 2024

A005117(a(n)) = A000040(n) = prime(n).
a(n) ~ (6/Pi^2) * n log n. - Charles R Greathouse IV, Nov 27 2017
a(n) = A013928(A008864(n)). - Ridouane Oudra, Oct 15 2019
From Gus Wiseman, Dec 08 2024: (Start)
a(n) = A112929(n) + 1.
a(n+1) - a(n) = A373198(n) = A061398(n) - 1.
(End)

A377783 Least nonsquarefree number > prime(n).

Original entry on oeis.org

4, 4, 8, 8, 12, 16, 18, 20, 24, 32, 32, 40, 44, 44, 48, 54, 60, 63, 68, 72, 75, 80, 84, 90, 98, 104, 104, 108, 112, 116, 128, 132, 140, 140, 150, 152, 160, 164, 168, 175, 180, 184, 192, 196, 198, 200, 212, 224, 228, 232, 234, 240, 242, 252, 260, 264, 270, 272

Offset: 1

Gus Wiseman, Nov 16 2024

nonn

No term appears more than twice. Proof: This would require at least 4 consecutive squarefree numbers (3 primes and at least 1 squarefree number between them). But we cannot have more than 3 consecutive squarefree numbers, because otherwise one of them must be divisible by 4, hence not squarefree.

			The third prime is 5, which is followed by 6, 7, 8, 9, ..., of which 8 is the first nonsquarefree term, so a(3) = 8.
The terms together with their prime indices begin:
    4: {1,1}
    4: {1,1}
    8: {1,1,1}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   32: {1,1,1,1,1}
   40: {1,1,1,3}
   44: {1,1,5}
   44: {1,1,5}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   60: {1,1,2,3}
   63: {2,2,4}
   68: {1,1,7}
   72: {1,1,1,2,2}

For squarefree we have A112926 (diffs A378037), opposite A112925 (diffs A378038).
Restriction to the primes of A120327, which has first differences A378039.
For prime-power instead of nonsquarefree (and primes + 1) we have A345531.
First differences are A377784.
The opposite is A378032 (diffs A378034), restriction of A378033 (diffs A378036).
The union is A378040.
Terms appearing only once are A378082.
Terms appearing twice are A378083.
Nonsquarefree numbers that are missing are A378084.
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.
A070321 gives the greatest squarefree number up to n.
Cf. A000015, A013597, A013928, A014210, A053797, A053806, A065514, A072284, A120992, A224363, A337030, A377047, A377048, A377049, A377430, A377431.

Table[NestWhile[#+1&,Prime[n],SquareFreeQ],{n,100}]

a(n) = A120327(prime(n)).

Proof suggested by Amiram Eldar.

A378037 First differences of A112926 (smallest squarefree integer > prime(n)).

Original entry on oeis.org

2, 1, 4, 3, 1, 5, 2, 5, 4, 3, 5, 4, 4, 5, 4, 6, 1, 7, 4, 1, 8, 3, 6, 10, 1, 3, 4, 1, 4, 15, 4, 5, 3, 10, 3, 4, 7, 5, 4, 7, 1, 11, 1, 5, 2, 12, 13, 3, 1, 5, 6, 5, 7, 5, 7, 6, 2, 5, 4, 3, 10, 14, 4, 1, 4, 16, 5, 10, 4, 1, 8, 8, 4, 7, 4, 5, 8, 4, 8, 11, 1, 11, 1

Offset: 1

Gus Wiseman, Dec 04 2024

nonn

First differences of A112926, restriction of A067535, differences A378087.
For prime powers we have A377703.
The nonsquarefree version is A377784 (differences of A377783), restriction of A378039.
The nonsquarefree opposite is A378034, first differences of A378032.
The opposite is A378038, differences of A112925.
The unrestricted opposite is A378085 except first term, differences of A070321.
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, A045882, A053797, A053806, A072284, A073247, A120327, A280892, A378036, A378040, A378084.

Differences[Table[NestWhile[#+1&,Prime[n]+1,!SquareFreeQ[#]&],{n,100}]]

A378038 First differences of A112925 = greatest squarefree number < prime(n).

Original entry on oeis.org

1, 1, 3, 4, 1, 4, 2, 5, 4, 4, 5, 4, 3, 4, 5, 7, 1, 7, 4, 1, 7, 4, 5, 8, 2, 5, 4, 1, 4, 12, 7, 4, 4, 8, 3, 6, 6, 5, 4, 8, 1, 11, 1, 4, 2, 13, 12, 4, 1, 4, 7, 1, 10, 6, 7, 5, 2, 5, 4, 4, 9, 14, 5, 1, 3, 16, 5, 11, 1, 2, 9, 8, 5, 6, 5, 4, 9, 4, 8, 11, 1, 11, 1, 7

Offset: 1

Gus Wiseman, Dec 04 2024

nonn

First differences of A112925, restriction of A070321, differences A378085.
For prime powers we have A377781, opposite A377703.
The nonsquarefree opposite is A377784 (differences of A377783), restriction of A378039.
The nonsquarefree version is A378034, first differences of A378032.
The opposite is A378037, differences of A112926.
The unrestricted opposite is A378087, differences of A067535.
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, A045882, A053797, A053806, A072284, A073247, A120327, A280892, A378036, A378040, A378084.

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

A377784 First-differences of A377783 (least nonsquarefree number > prime(n)).

Original entry on oeis.org

0, 4, 0, 4, 4, 2, 2, 4, 8, 0, 8, 4, 0, 4, 6, 6, 3, 5, 4, 3, 5, 4, 6, 8, 6, 0, 4, 4, 4, 12, 4, 8, 0, 10, 2, 8, 4, 4, 7, 5, 4, 8, 4, 2, 2, 12, 12, 4, 4, 2, 6, 2, 10, 8, 4, 6, 2, 7, 5, 0, 10, 14, 4, 3, 5, 12, 6, 10, 2, 6, 4, 8, 7, 5, 4, 8, 8, 4, 8, 8, 3, 9, 4, 4

Offset: 1

Gus Wiseman, Nov 18 2024

nonn

There are no consecutive 0's.

Does this sequence contain every positive integer > 1?

Positions of 0's are A068361.
The opposite for squarefree is A378038, differences of A112925.
For prime-power instead of nonsquarefree and primes + 1 we have A377703, first-differences of A345531.
First-differences of A377783, union A378040.
The opposite is A378034 (differences of A378032), restriction of A378036 (differences A378033).
For squarefree instead of nonsquarefree we have A378037, first-differences of A112926.
Restriction of A378039 (first-differences of A120327) to the primes.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A061398, A068360, A337030, A377430, A377431 count squarefree numbers between primes.
A061399, A068361, A378086 count nonsquarefree numbers between primes.
A070321 gives the greatest squarefree number up to n.
Cf. A000015, A013928, A053797, A053806, A072284, A224363.
Cf. A377047, A377048, A377049.
Cf. A378082, A378083, A378084.

Differences[Table[NestWhile[#+1&,Prime[n],SquareFreeQ[#]&],{n,100}]]

A378039 a(1)=3; a(n>1) = n-th first difference of A120327(k) = least nonsquarefree number greater than k.

Original entry on oeis.org

3, 0, 0, 4, 0, 0, 0, 1, 3, 0, 0, 4, 0, 0, 0, 2, 0, 2, 0, 4, 0, 0, 0, 1, 2, 0, 1, 4, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 1, 3, 0, 0, 1, 1, 2, 0, 2, 0, 2, 0, 4, 0, 0, 0, 3, 0, 0, 1, 4, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 1, 4, 0, 0, 0, 1, 3, 0, 0, 4, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 18 2024

nonn

The union is {0,1,2,3,4}.

Positions of 0's are A005117.
Positions of 4's are A007675 - 1, except first term.
Positions of 1's are A068781.
Positions of 2's are A073247 - 1.
Positions of 3's are A073248 - 1, except first term.
First-differences of A120327.
For prime-powers we have A377780, first-differences of A000015.
Restriction is A377784 (first-differences of A377783, union A378040).
The opposite is A378036 (differences A378033), for prime-powers A377782.
The opposite for squarefree is A378085, differences of A070321
For squarefree we have A378087, restriction A378037, differences of A112926.
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. A013928, A053797, A053806, A072284, A224363, A377047, A377049, A378032, A378034, A378082, A378083, A378084.

Differences[Table[NestWhile[#+1&,n,#>1&&SquareFreeQ[#]&],{n,100}]]

A378086 Number of nonsquarefree numbers < prime(n).

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 5, 6, 7, 11, 11, 13, 14, 14, 16, 20, 22, 23, 25, 26, 27, 29, 31, 33, 36, 39, 39, 40, 41, 42, 49, 50, 53, 53, 57, 58, 61, 63, 64, 68, 70, 71, 74, 75, 76, 77, 81, 84, 86, 87, 88, 90, 91, 97, 99, 101, 103, 104, 107, 109, 109, 113, 119, 120, 121

Offset: 1

Gus Wiseman, Dec 04 2024

nonn

			The nonsquarefree numbers counted under each term begin:
  n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9: n=10: n=11: n=12:
  --------------------------------------------------------------
   .    .    4    4    9    12   16   18   20   28    28    36
                       8    9    12   16   18   27    27    32
                       4    8    9    12   16   25    25    28
                            4    8    9    12   24    24    27
                                 4    8    9    20    20    25
                                      4    8    18    18    24
                                           4    16    16    20
                                                12    12    18
                                                9     9     16
                                                8     8     12
                                                4     4     9
                                                            8
                                                            4

For nonprime numbers we have A014689.
Restriction of A057627 to the primes.
First-differences are A061399 (zeros A068361), squarefree A061398 (zeros A068360).
For composite instead of squarefree we have A065890.
For squarefree we have A071403, differences A373198.
Greatest is A378032 (differences A378034), restriction of A378033 (differences A378036).
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A070321 gives the greatest squarefree number up to n.
A112925 gives the greatest squarefree number between primes, differences A378038.
A112926 gives the least squarefree number between primes, differences A378037.
A120327 gives the least nonsquarefree number >= n, first-differences A378039.
A377783 gives the least nonsquarefree > prime(n), differences A377784.
Cf. A013928, A046933, A053797, A053806, A072284, A076259, A224363, A337030, A377049, A378040, A378082, A378084.

Table[Length[Select[Range[Prime[n]],!SquareFreeQ[#]&]],{n,100}]

from math import isqrt
from sympy import prime, mobius
def A378086(n): return (p:=prime(n))-sum(mobius(k)*(p//k**2) for k in range(1,isqrt(p)+1)) # Chai Wah Wu, Dec 05 2024

a(n) = A057627(prime(n)).

A071172 Number of squarefree integers <= 10^n.

Original entry on oeis.org

1, 7, 61, 608, 6083, 60794, 607926, 6079291, 60792694, 607927124, 6079270942, 60792710280, 607927102274, 6079271018294, 60792710185947, 607927101854103, 6079271018540405, 60792710185403794, 607927101854022750, 6079271018540280875, 60792710185402613302, 607927101854026645617

Offset: 0

Robert G. Wilson v, Jun 10 2002

nonn

The limit of a(n)/10^n is 6/Pi^2 (see A059956). - Gerard P. Michon, Apr 30 2009

J. Pawlewicz, Table of n, a(n) for n = 0..36
W. Hürlimann, A First Digit Theorem for Square-Free Integer Powers, Pure Mathematical Sciences, Vol. 3, 2014, no. 3, 129 - 139 HIKARI Ltd.
G. P. Michon, On the number of squarefree integers not exceeding N. - _Gerard P. Michon_, Apr 30 2009
J. Pawlewicz, Counting square-free numbers, arXiv preprint arXiv:1107.4890 [math.NT], 2011.
Eric Weisstein's World of Mathematics, Squarefree

Cf. A005117, A013928.
Apart from first two terms, same as A053462.
Binary counterpart is A143658. - Gerard P. Michon, Apr 30 2009

f[n_] := Sum[ MoebiusMu[i]Floor[n/i^2], {i, Sqrt@ n}]; Table[ f[10^n], {n, 0, 14}] (* Robert G. Wilson v, Aug 04 2012 *)

a(n)=sum(d=1,sqrtint(n=10^n),moebius(d)*n\d^2) \\ Charles R Greathouse IV, Nov 14 2012

a(n)=my(s); forsquarefree(d=1,sqrtint(n=10^n), s += n\d[1]^2 * moebius(d)); s \\ Charles R Greathouse IV, Jan 08 2018

from math import isqrt
from sympy import mobius
def A071172(n): return sum(mobius(k)*(10**n//k**2) for k in range(1,isqrt(10**n)+1)) # Chai Wah Wu, May 10 2024

a(n) = Sum_{i=1..10^(n/2)} A008683(i)*floor(10^n/i^2). - Gerard P. Michon, Apr 30 2009

Extended by Eric W. Weisstein, Sep 14 2003
3 more terms from Jud McCranie, Sep 01 2005
4 more terms from Gerard P. Michon, Apr 30 2009

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

A378040 Union of A377783(n) = least nonsquarefree number > prime(n).

Original entry on oeis.org

4, 8, 12, 16, 18, 20, 24, 32, 40, 44, 48, 54, 60, 63, 68, 72, 75, 80, 84, 90, 98, 104, 108, 112, 116, 128, 132, 140, 150, 152, 160, 164, 168, 175, 180, 184, 192, 196, 198, 200, 212, 224, 228, 232, 234, 240, 242, 252, 260, 264, 270, 272, 279, 284, 294, 308, 312

Offset: 1

Gus Wiseman, Nov 20 2024

nonn

Numbers k such that, if p is the greatest prime < k, all numbers from p to k (exclusive) are squarefree.

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

For squarefree we have A112926 (diffs A378037), opposite A112925 (diffs A378038).
For prime-power instead of nonsquarefree we have A345531, differences A377703.
Union of A377783 (diffs A377784), restriction of A120327 (diffs A378039).
Nonsquarefree numbers not appearing are A378084, see also A378082, A378083.
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.
A070321 gives the greatest squarefree number up to n.
A071403(n) = A013928(prime(n)) counts squarefree numbers up to prime(n).
A378086(n) = A057627(prime(n)) counts nonsquarefree numbers up to prime(n).
Cf. A378034 (differences of A378032), restriction of A378036 (differences A378033).
Cf. A000015, A053797, A053806, A072284, A224363, A337030, A377047, A377049, A377430, A377431.

Union[Table[NestWhile[#+1&,Prime[n],SquareFreeQ],{n,100}]]
lns[p_]:=Module[{k=p+1},While[SquareFreeQ[k],k++];k]; Table[lns[p],{p,Prime[Range[70]]}]//Union (* Harvey P. Dale, Jun 12 2025 *)

cp's OEIS Frontend

A071403 Which squarefree number is prime? a(n)-th squarefree number equals n-th prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

Python

Formula

A377783 Least nonsquarefree number > prime(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A378037 First differences of A112926 (smallest squarefree integer > prime(n)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A378038 First differences of A112925 = greatest squarefree number < prime(n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A377784 First-differences of A377783 (least nonsquarefree number > prime(n)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A378039 a(1)=3; a(n>1) = n-th first difference of A120327(k) = least nonsquarefree number greater than k.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A378086 Number of nonsquarefree numbers < prime(n).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Python

Formula

A071172 Number of squarefree integers <= 10^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python