A036263 -id:A036263 - 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.

A377038 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the squarefree numbers.

Original entry on oeis.org

1, 2, 1, 3, 1, 0, 5, 2, 1, 1, 6, 1, -1, -2, -3, 7, 1, 0, 1, 3, 6, 10, 3, 2, 2, 1, -2, -8, 11, 1, -2, -4, -6, -7, -5, 3, 13, 2, 1, 3, 7, 13, 20, 25, 22, 14, 1, -1, -2, -5, -12, -25, -45, -70, -92, 15, 1, 0, 1, 3, 8, 20, 45, 90, 160, 252, 17, 2, 1, 1, 0, -3, -11, -31, -76, -166, -326, -578

Offset: 0

Gus Wiseman, Oct 18 2024

Row n is the k-th differences of A005117 = the squarefree numbers.

			Array form:
        n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
  ----------------------------------------------------------
  k=0:   1     2     3     5     6     7    10    11    13
  k=1:   1     1     2     1     1     3     1     2     1
  k=2:   0     1    -1     0     2    -2     1    -1     0
  k=3:   1    -2     1     2    -4     3    -2     1     1
  k=4:  -3     3     1    -6     7    -5     3     0    -2
  k=5:   6    -2    -7    13   -12     8    -3    -2     3
  k=6:  -8    -5    20   -25    20   -11     1     5    -5
  k=7:   3    25   -45    45   -31    12     4   -10    10
  k=8:  22   -70    90   -76    43    -8   -14    20   -19
  k=9: -92   160  -166   119   -51    -6    34   -39    28
Triangle form:
   1
   2   1
   3   1   0
   5   2   1   1
   6   1  -1  -2  -3
   7   1   0   1   3   6
  10   3   2   2   1  -2  -8
  11   1  -2  -4  -6  -7  -5   3
  13   2   1   3   7  13  20  25  22
  14   1  -1  -2  -5 -12 -25 -45 -70 -92
  15   1   0   1   3   8  20  45  90 160 252

Row k=0 is A005117.
Row k=1 is A076259.
Row k=2 is A376590.
The version for primes is A095195, noncomposites A376682, composites A377033.
A version for partitions is A175804, cf. A053445, A281425, A320590.
Triangle row-sums are A377039, absolute version A377040.
Column n = 1 is A377041, for primes A007442 or A030016.
First position of 0 in each row is A377042.
For nonsquarefree instead of squarefree numbers we have A377046.
For prime-powers instead of squarefree numbers we have A377051.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, complement A013929 (differences A078147).
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
Cf. A007674, A053797, A053806, A061398, A072284, A075526, A112925, A112926, A120992, A373198, A376311, A376591, A376592.

nn=9;
t=Table[Take[Differences[NestList[NestWhile[#+1&,#+1,!SquareFreeQ[#]&]&,1,2*nn],k],nn],{k,0,nn}]
Table[t[[j,i-j+1]],{i,nn},{j,i}]

A(i,j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) A005117(i+k).

A377430 Numbers k such that there is exactly one squarefree number between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

3, 4, 9, 10, 13, 14, 15, 22, 26, 33, 39, 48, 59, 60, 65, 85, 88, 89, 93, 104, 113, 116, 122, 142, 143, 147, 148, 155, 181, 188, 198, 201, 209, 212, 213, 224, 226, 234, 235, 244, 254, 264, 265, 268, 287, 288, 313, 320, 328, 332, 333, 341, 343, 353, 361, 366

Offset: 1

Gus Wiseman, Oct 29 2024

nonn

			Primes 4 and 5 are 7 and 11, and the interval (8,9,10) contains only squarefree 10, so 4 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

For composite instead of squarefree we have A029707.
These are the positions of 1 in A061398, or 2 in A373198.
For no squarefree numbers we have A068360.
For prime-power instead of squarefree we have A377287.
For at least one squarefree number we have A377431.
For perfect-power instead of squarefree we have A377434.
A000040 lists the primes, differences A001223, seconds A036263.
A002808 lists the composites, complement A008578.
A005117 lists the squarefree numbers, complement A013929.
A377038 gives k-differences of squarefree numbers.
Cf. A007674, A013603, A036378, A046933, A072284, A076259, A077641, A077643, A075526, A078147, A112925, A112926, A120992, A293697.

R:= NULL: count:= 0: q:= 2:
for k from 1 while count < 100 do
  p:= q; q:= nextprime(q);
  if nops(select(numtheory:-issqrfree,[$p+1 .. q-1]))=1 then
    R:= R,k; count:= count+1;
 fi
od:
R; # Robert Israel, Nov 29 2024

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],SquareFreeQ]]==1&]

is(n,p=prime(n))=my(q=nextprime(p+1),s); for(k=p+1,q-1, if(issquarefree(k) && s++>1, return(0))); s==1 \\ Charles R Greathouse IV, Nov 29 2024

A376597 Inflection and undulation points in the sequence of prime-powers inclusive (A000961).

Original entry on oeis.org

1, 2, 3, 6, 8, 14, 15, 16, 27, 32, 50, 61, 67, 72, 85, 92, 93, 124, 129, 132, 136, 141, 185, 190, 211, 214, 221, 226, 268, 292, 301, 302, 322, 374, 394, 423, 456, 463, 502, 503, 547, 559, 560, 593, 604, 640, 646, 663, 671, 675, 710, 726, 727, 746, 754, 755

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A376596) are zero.

Inclusive means 1 is a prime-power. For the exclusive version, subtract 1 and shift left.

			The prime-powers inclusive (A000961) are:
  1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
with first differences (A057820):
  1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, ...
with first differences (A376596):
  0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...
with zeros (A376597) at:
  1, 2, 3, 6, 8, 14, 15, 16, 27, 32, 50, 61, 67, 72, 85, 92, 93, 124, 129, 132, ...

Gus Wiseman, Inflection and undulation points in the prime-powers inclusive.

The first differences were A057820, see also A053707, A376340.
These are the zeros of A376596 (sorted firsts A376653, exclusive A376654).
The complement is A376598.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
A064113 lists positions of adjacent equal prime gaps.
For prime-powers inclusive: A057820 (first differences), A376596 (second differences), A376598 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
Cf. A025475, A053707, A069623, A093555, A174965, A361102, A376653, A376654.

Join@@Position[Differences[Select[Range[1000],#==1||PrimePowerQ[#]&],2],0]

A377033 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the composite numbers (A002808).

Original entry on oeis.org

4, 6, 2, 8, 2, 0, 9, 1, -1, -1, 10, 1, 0, 1, 2, 12, 2, 1, 1, 0, -2, 14, 2, 0, -1, -2, -2, 0, 15, 1, -1, -1, 0, 2, 4, 4, 16, 1, 0, 1, 2, 2, 0, -4, -8, 18, 2, 1, 1, 0, -2, -4, -4, 0, 8, 20, 2, 0, -1, -2, -2, 0, 4, 8, 8, 0, 21, 1, -1, -1, 0, 2, 4, 4, 0, -8, -16, -16

Offset: 0

Gus Wiseman, Oct 17 2024

Row n is the k-th differences of A002808 = the composite numbers.

			Array begins:
        n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
  ----------------------------------------------------------
  k=0:   4     6     8     9    10    12    14    15    16
  k=1:   2     2     1     1     2     2     1     1     2
  k=2:   0    -1     0     1     0    -1     0     1     0
  k=3:  -1     1     1    -1    -1     1     1    -1    -1
  k=4:   2     0    -2     0     2     0    -2     0     2
  k=5:  -2    -2     2     2    -2    -2     2     2    -2
  k=6:   0     4     0    -4     0     4     0    -4    -1
  k=7:   4    -4    -4     4     4    -4    -4     3    10
  k=8:  -8     0     8     0    -8     0     7     7   -29
  k=9:   8     8    -8    -8     8     7     0   -36    63
Triangle begins:
    4
    6    2
    8    2    0
    9    1   -1   -1
   10    1    0    1    2
   12    2    1    1    0   -2
   14    2    0   -1   -2   -2    0
   15    1   -1   -1    0    2    4    4
   16    1    0    1    2    2    0   -4   -8
   18    2    1    1    0   -2   -4   -4    0    8
   20    2    0   -1   -2   -2    0    4    8    8    0
   21    1   -1   -1    0    2    4    4    0   -8  -16  -16

Initial rows: A002808, A073783, A073445.
The version for primes is A095195 or A376682.
A version for partitions is A175804, cf. A053445, A281425, A320590.
Triangle row-sums are A377034, absolute version A377035.
Column n = 1 is A377036, for primes A007442 or A030016.
First position of 0 in each row is A377037.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, seconds A036263.
A008578 lists the noncomposites, differences A075526.
Cf. A065310, A065890, A084758, A173390, A350004, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680.

nn=9;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,PrimeQ]&,4,2*nn],k],nn],{k,0,nn}]

A(i,j) = Sum_{k=0..j} (-1)^(j-k) binomial(j,k) A002808(i+k).

A377049 First term of the n-th differences of the nonsquarefree numbers. Inverse zero-based binomial transform of A013929.

Original entry on oeis.org

4, 4, -3, 5, -6, 4, 3, -15, 25, -10, -84, 369, -1067, 2610, -5824, 12246, -24622, 47577, -88233, 155962, -259086, 393455, -512281, 456609, 191219, -2396571, 8213890, -21761143, 50923029, -110269263, 225991429, -444168664, 844390152, -1561482492, 2817844569

Offset: 0

Gus Wiseman, Oct 19 2024

sign

Harvey P. Dale, Table of n, a(n) for n = 0..400

The version for primes is A007442, noncomposites A030016, composites A377036.
For squarefree instead of nonsquarefree numbers we have A377041.
For antidiagonal-sums we have A377047, absolute A377048.
For first position of 0 in each row we have A377050.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
Cf. A007674, A053797, A053806, A061398, A072284, A075526, A084758, A112925, A120992, A251092, A376311, A376591.

nn=20;
Table[First[Differences[NestList[NestWhile[#+1&, #+1,SquareFreeQ[#]&]&,4,2*nn],k]],{k,0,nn}]
With[{nsf=Select[Range[1000],!SquareFreeQ[#]&]},Table[Differences[nsf,n],{n,0,40}]][[;;,1]] (* Harvey P. Dale, Nov 28 2024 *)

The inverse zero-based binomial transform of a sequence (q(0), q(1), q(2), ...) is the sequence p given by:

p(j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) q(k)

A377431 Numbers k such that there is at least one squarefree number between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

3, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 42, 44, 46, 47, 48, 50, 51, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 84, 85, 86

Offset: 1

Gus Wiseman, Oct 29 2024

nonn

			Primes 4 and 5 are 7 and 11, and the interval (8,9,10) contains 10, which is squarefree, so 4 is in the sequence.

These are the positive positions in A061398, or terms >= 2 in A373198.
The complement (no squarefree numbers) is A068360.
For prime-power instead of squarefree we have A377057, strict version A377287.
For exactly one squarefree number we have A377430.
A000040 lists the primes, differences A001223, seconds A036263.
A002808 lists the composites, complement A008578.
A005117 lists the squarefree numbers, complement A013929.
A377038 gives k-differences of squarefree numbers.
Cf. A007674, A013603, A029707, A036378, A046933, A072284, A076259, A077641, A077643, A075526, A078147, A112925, A112926, A120992, A293697.

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],SquareFreeQ]]>=1&]

A378036 First differences of A378033 (greatest positive integer < n that is 1 or nonsquarefree).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 18 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65539

Positions of 0 are A005117 - 1, complement A013929 - 1.
Sums for squarefree numbers are A070321 (restriction A112925).
The restricted opposite is A377784, differences of A377783 (union A378040).
First-differences of A378033.
The restriction is A378034, differences of A378032.
The restricted opposite for squarefree is A378037, differences of A112926.
The opposite is A378039, differences of A120327 (union A162966).
For squarefree numbers we have A378085, restriction A378038.
The opposite for squarefree is A378087, differences of A067535.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259, seconds A376590.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A061398 counts squarefree numbers between primes (sums A337030), zeros A068360.
A061399 counts nonsquarefree numbers between primes (sums A378086), zeros A068361.
A377046 encodes k-differences of nonsquarefree numbers, zeros A377050.
Cf. A053797, A053806, A065514, A377047, A377048, A377049, A377703, A378082, A378083, A378084.

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

A378033(n) = if(n<=3, 1, forstep(k=n, 0, -1, if(!issquarefree(k), return(k))));
A378036(n) = (A378033(1+n)-A378033(n)); \\ Antti Karttunen, Jan 28 2025

a(prime(n)) = A378034(n).

Data section extended to a(107) by Antti Karttunen, Jan 28 2025

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}]]

cp's OEIS Frontend

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

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A377783 Least nonsquarefree number > prime(n).

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A377038 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the squarefree numbers.

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A377430 Numbers k such that there is exactly one squarefree number between prime(k)+1 and prime(k+1)-1.

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A376597 Inflection and undulation points in the sequence of prime-powers inclusive (A000961).

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A377033 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the composite numbers (A002808).

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A377049 First term of the n-th differences of the nonsquarefree numbers. Inverse zero-based binomial transform of A013929.

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A377431 Numbers k such that there is at least one squarefree number between prime(k)+1 and prime(k+1)-1.

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