A378039 -id:A378039 - cp's OEIS frontend

A378086 Number of nonsquarefree numbers < prime(n).

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)).

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 *)

A378084 Nonsquarefree numbers not appearing in A377783 (least nonsquarefree number > prime(n)).

Original entry on oeis.org

9, 25, 27, 28, 36, 45, 49, 50, 52, 56, 64, 76, 81, 88, 92, 96, 99, 100, 117, 120, 121, 124, 125, 126, 135, 136, 144, 147, 148, 153, 156, 162, 169, 171, 172, 176, 188, 189, 204, 207, 208, 216, 220, 225, 236, 243, 244, 245, 248, 250, 256, 261, 268, 275, 276, 280

Offset: 1

Gus Wiseman, Nov 23 2024

nonn

Warning: do not confuse with A377784.

			The terms together with their prime indices begin:
    9: {2,2}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   36: {1,1,2,2}
   45: {2,2,3}
   49: {4,4}
   50: {1,3,3}
   52: {1,1,6}
   56: {1,1,1,4}
   64: {1,1,1,1,1,1}
   76: {1,1,8}
   81: {2,2,2,2}
   88: {1,1,1,5}
   92: {1,1,9}
   96: {1,1,1,1,1,2}

Disjoint from A377783 (union A378040), first-differences A377784.
Appearing once: A378082.
Appearing twice: 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 (sums A337030), zeros A068360.
A061399 counts nonsquarefree numbers between primes (sums A378086), zeros A068361.
A070321 gives the greatest squarefree number up to n.
A112925 gives least squarefree number > prime(n), differences A378038.
A112926 gives greatest squarefree number < prime(n), differences A378037.
A120327 (union A162966) gives least nonsquarefree number >= n, differences A378039.
A377046 encodes k-differences of nonsquarefree numbers, zeros A377050.
Cf. A053797, A053806, A072284, A224363, A377430, A377431, A378032, A378033.

nn=100;
y=Table[NestWhile[#+1&,Prime[n],SquareFreeQ[#]&],{n,nn}];
Complement[Select[Range[Prime[nn]],!SquareFreeQ[#]&],y]

Complement of A378040 in A013929.

A378082 Terms appearing only once in A377783 = least nonsquarefree number > prime(n).

Original entry on oeis.org

12, 16, 18, 20, 24, 40, 48, 54, 60, 63, 68, 72, 75, 80, 84, 90, 98, 108, 112, 116, 128, 132, 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, 294, 308, 312, 315, 320, 332, 338, 348

Offset: 1

Gus Wiseman, Nov 20 2024

nonn

Nonsquarefree numbers k such that if p < q are the two greatest primes < k, there is at least one nonsquarefree number between p and q but all numbers between q and k are squarefree. - Robert Israel, Nov 20 2024

			The terms together with their prime indices begin:
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   40: {1,1,1,3}
   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}
   75: {2,3,3}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   90: {1,2,2,3}
   98: {1,4,4}
  108: {1,1,2,2,2}
  112: {1,1,1,1,4}
  116: {1,1,10}
  128: {1,1,1,1,1,1,1}
  132: {1,1,2,5}

This is a transformation of A377783 (union A378040, differences A377784).
Note also A377783 restricts A120327 (differences A378039) to the primes.
Terms appearing twice are A378083.
Terms not appearing at all 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.
A071403(n) = A013928(prime(n)) counts squarefree numbers < prime(n).
A378086(n) = A057627(prime(n)) counts nonsquarefree numbers < prime(n).
Cf. A112926 (diffs A378037), opposite A112925 (diffs A378038).
Cf. A378032 (diffs A378034), restriction of A378033 (diffs A378036).
Cf. A053797, A053806, A072284, A112929, A120992, A224363, A337030, A377430, A377431, A377703.

q:= 3: R:= NULL: flag:= false: count:= 0:
while count < 100 do
  p:= q; q:= nextprime(q);
  for k from p+1 to q-1 do
    found:= false;
    if not numtheory:-issqrfree(k) then
      if flag then
          count:= count+1; R:= R,k
      fi;
      found:= true; break
    fi;
   od;
   flag:= found;
od:
R; # Robert Israel, Nov 20 2024

y=Table[NestWhile[#+1&,Prime[n],SquareFreeQ],{n,100}];
Select[Most[Union[y]],Count[y,#]==1&]

A378085 First differences of A070321 (greatest squarefree number <= n).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 04 2024

nonn

			The greatest squarefree number <= 50 is 47, and the greatest squarefree number <= 51 is 51, so a(51) = 4.

Ones are A007674.
Zeros are A013929 - 1.
Twos are A280892.
Positions of first appearances are A020755 - 1 (except first term).
First-differences of A070321.
The nonsquarefree restriction is A378034, differences of A378032.
For nonsquarefree numbers we have A378036, differences of A378033.
The opposite restriction to primes is A378037, differences of A112926.
The restriction to primes is A378038, differences of A112925.
The nonsquarefree opposite is A378039, restriction A377784.
The opposite version is A378087.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
Cf. A013928, A020754, A045882, A053797, A053806, A067535, A068781, A072284, A073247, A073248, A120327.

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

A378083 Nonsquarefree numbers appearing exactly twice in A377783 (least nonsquarefree number > prime(n)).

Original entry on oeis.org

4, 8, 32, 44, 104, 140, 284, 464, 572, 620, 644, 824, 860, 1232, 1292, 1304, 1484, 1700, 1724, 1880, 2084, 2132, 2240, 2312, 2384, 2660, 2732, 2804, 3392, 3464, 3560, 3920, 3932, 4004, 4220, 4244, 4424, 4640, 4724, 5012, 5444, 5480, 5504, 5660, 6092, 6200

Offset: 1

Gus Wiseman, Nov 23 2024

nonn

Warning: do not confuse with A377783.

			The terms together with their prime indices begin:
     4: {1,1}
     8: {1,1,1}
    32: {1,1,1,1,1}
    44: {1,1,5}
   104: {1,1,1,6}
   140: {1,1,3,4}
   284: {1,1,20}
   464: {1,1,1,1,10}
   572: {1,1,5,6}
   620: {1,1,3,11}
   644: {1,1,4,9}
   824: {1,1,1,27}
   860: {1,1,3,14}
  1232: {1,1,1,1,4,5}

Subset of A377783 (union A378040, diffs A377784), restriction of A120327 (diffs A378039).
Terms appearing once are A378082.
Terms not appearing at all are A378084.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
A071403(n) = A013928(prime(n)) counts squarefree numbers < prime(n).
A378086(n) = A057627(prime(n)) counts nonsquarefree numbers < prime(n).
Cf. A112926 (diffs A378037), opposite A112925 (diffs A378038).
Cf. A378032 (diffs A378034), restriction of A378033 (diffs A378036).
Cf. A053797, A053806, A070321, A072284, A112929, A120992, A224363, A337030, A377430, A377431, A377703.

y=Table[NestWhile[#+1&,Prime[n],SquareFreeQ[#]&],{n,1000}];
Select[Union[y],Count[y,#]==2&]

A378087 First-differences of A067535 (least positive integer >= n that is squarefree).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 20 2024

nonn

Does this contain all nonnegative integers? The positions of first appearances begin: 4, 1, 3, 7, 47, 241, 843, 22019, 217069, ...

Ones are A007674.
Zeros are A013929, complement A005117.
Positions of first appearances are A020754 (except first term) = A045882 - 1.
First-differences of A067535.
Twos are A280892.
For prime-powers we have A377780, differences of A000015.
The nonsquarefree opposite is A378036, differences of A378033.
The restriction to primes + 1 is A378037 (opposite A378038), differences of A112926.
For nonsquarefree numbers we have A378039, see A377783, A377784, A378040.
The opposite is A378085, 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. A005117, A007675, A068781, A073247, A073248, A378084.
Cf. A013928, A053797, A053806, A072284, A120327, A224363.

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

A377782 First-differences of A031218(n) = greatest number <= n that is 1 or a prime-power.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 16 2024

nonn

Note 1 is a power of a prime (A000961) but not a prime-power (A246655).

Positions of 1 are A006549.
Positions of 0 are A080765 = A024619 - 1, complement A181062 = A000961 - 1.
Positions of 2 are A120432 (except initial terms).
Sorted positions of first appearances appear to include A167236 - 1.
Positions of terms > 1 are A373677.
The restriction to primes minus 1 is A377289.
Below, A (B) indicates that A is the first-differences of B:
- This sequence is A377782 (A031218), which has restriction to primes A065514 (A377781).
- The opposite is A377780 (A000015), restriction A377703 (A345531).
- For nonsquarefree we have A378036 (A378033), opposite A378039 (A120327).
- For squarefree we have A378085 (A112925), restriction A378038 (A070321).
A000040 lists the primes, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A361102 lists the non-powers of primes, differences A375708.
A378034 gives differences of A378032 (restriction of A378033).
Prime-powers between primes: A053607, A080101, A366833, A377057, A377286, A377287.
Cf. A053289, A343249, A375706, A377281, A377282.

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

A378373 Number of composite numbers (A002808) between consecutive nonsquarefree numbers (A013929), exclusive.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 02 2024

nonn

All terms are 0, 1, 2, or 3 (cf. A078147).
The inclusive version is a(n) + 2.
The nonsquarefree numbers begin: 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, ...

			The composite numbers counted by a(n) form the following set partition of A120944:
{6}, {}, {10}, {14,15}, {}, {}, {21,22}, {}, {26}, {}, {30}, {33,34,35}, {38,39}, ...

For prime (instead of nonsquarefree) we have A046933.
For squarefree (instead of nonsquarefree) we have A076259(n)-1.
For prime power (instead of nonsquarefree) we have A093555.
For prime instead of composite we have A236575.
For nonprime prime power (instead of nonsquarefree) we have A378456.
For perfect power (instead of nonsquarefree) we have A378614, primes A080769.
A002808 lists the composite numbers.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A073247 lists squarefree numbers with nonsquarefree neighbors.
A120944 lists squarefree composite numbers.
A377432 counts perfect-powers between primes, zeros A377436.
A378369 gives distance to the next nonsquarefree number (A120327).
Cf. A065890, A067535, A067871, A080101, A081221, A151800, A243348, A366833, A377046, A378033, A378039, A378086.

v=Select[Range[100],!SquareFreeQ[#]&];
Table[Length[Select[Range[v[[i]]+1,v[[i+1]]-1],CompositeQ]],{i,Length[v]-1}]

A378369 Distance between n and the least nonsquarefree number >= n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 01 2024

nonn

All terms are 0, 1, 2, or 3 (cf. A078147).

Adding n to each term a(n) gives A120327.
Positions of 0 are A013929.
Positions of 1 are A373415.
Positions of 2 are A378458.
Positions of 3 are A007675.
Sequences obtained by adding n to each term are placed in parentheses below.
The version for primes is A007920 (A007918).
The version for perfect powers is A074984 (A377468).
The version for squarefree numbers is A081221 (A067535).
The version for non-perfect powers is A378357 (A378358).
The version for prime powers is A378370 (A000015).
The version for non prime powers is A378371 (A378372).
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A120992 gives run-lengths of squarefree numbers increasing by one.
Cf. A073247, A151800, A236575, A243348, A375707, A377046, A378033, A378039, A378086, A378373.

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

cp's OEIS Frontend

A378086 Number of nonsquarefree numbers < prime(n).

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

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A378084 Nonsquarefree numbers not appearing in A377783 (least nonsquarefree number > prime(n)).

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A378082 Terms appearing only once in A377783 = least nonsquarefree number > prime(n).

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A378085 First differences of A070321 (greatest squarefree number <= n).

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A378083 Nonsquarefree numbers appearing exactly twice in A377783 (least nonsquarefree number > prime(n)).

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A378087 First-differences of A067535 (least positive integer >= n that is squarefree).

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A377782 First-differences of A031218(n) = greatest number <= n that is 1 or a prime-power.

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A378373 Number of composite numbers (A002808) between consecutive nonsquarefree numbers (A013929), exclusive.

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