A377049 -id:A377049 - cp's OEIS frontend

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

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

A378033 Greatest nonsquarefree number <= n, or 1 if there is none (the case n <= 3).

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 4, 8, 9, 9, 9, 12, 12, 12, 12, 16, 16, 18, 18, 20, 20, 20, 20, 24, 25, 25, 27, 28, 28, 28, 28, 32, 32, 32, 32, 36, 36, 36, 36, 40, 40, 40, 40, 44, 45, 45, 45, 48, 49, 50, 50, 52, 52, 54, 54, 56, 56, 56, 56, 60, 60, 60, 63, 64, 64, 64, 64, 68

Offset: 1

Gus Wiseman, Nov 18 2024

nonn

			The nonsquarefree numbers <= 10 are {4, 8, 9}, so a(10) = 9.

For prime-powers we have A031218, differences A377782.
Greatest of the nonsquarefree numbers counted by A057627.
The opposite for squarefree is A067535, differences A378087.
For squarefree we have A070321, differences A378085.
The opposite is A120327 (union A162966), differences A378039.
The restriction to the primes is A378032, opposite A377783 (union A378040).
First-differences are A378036, restriction A378034.
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.
A112925 gives the greatest squarefree number < prime(n), differences A378038.
A112926 gives the least squarefree number > prime(n), differences A378037.
A377046 encodes k-differences of nonsquarefree numbers, zeros A377050.
Cf. A053797, A053806, A377047, A377048, A377049, A377784, A378082, A378083, A378084.

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

a(n) = my(k=n); while (issquarefree(k), k--); if(!k, 1, k); \\ Michel Marcus, Jul 26 2025

a(prime(n)) = A378032(n).

a(n) = A013929(A057627(n)), for n > 3. - Ridouane Oudra, Jul 26 2025

A378032 a(1) = a(2) = 1; a(n>2) is the greatest nonsquarefree number < prime(n).

Original entry on oeis.org

1, 1, 4, 4, 9, 12, 16, 18, 20, 28, 28, 36, 40, 40, 45, 52, 56, 60, 64, 68, 72, 76, 81, 88, 96, 100, 100, 104, 108, 112, 126, 128, 136, 136, 148, 150, 156, 162, 164, 172, 176, 180, 189, 192, 196, 198, 208, 220, 225, 228, 232, 236, 240, 250, 256, 261, 268, 270

Offset: 1

Gus Wiseman, Nov 16 2024

nonn

			The terms together with their prime indices begin:
    1: {}
    1: {}
    4: {1,1}
    4: {1,1}
    9: {2,2}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   28: {1,1,4}
   28: {1,1,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   40: {1,1,1,3}
   45: {2,2,3}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}
   68: {1,1,7}
   72: {1,1,1,2,2}

Terms appearing twice are A061351 + 1.
For prime-powers we have A065514 (diffs A377781), opposite A345531 (diffs A377703).
For squarefree we have A112925 (differences A378038).
The opposite for squarefree is A112926 (differences A378037).
The opposite is A377783 (union A378040), restriction of A120327 (differences A378039).
Restriction of A378033, which has differences A378036.
The first-differences are A378034, opposite A377784.
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.
A377046 encodes k-differences of nonsquarefree numbers, zeros A377050.
Cf. A053797, A053806, A072284, A065890, A224363, A366833, A377431.
Cf. A377047, A377048, A377049, A377430, A378082, A378083, A378084.

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

a(n) = A378033(prime(n)).

A377046 Array read by downward antidiagonals where A(n,k) is the n-th term of the k-th differences of nonsquarefree numbers.

Original entry on oeis.org

4, 8, 4, 9, 1, -3, 12, 3, 2, 5, 16, 4, 1, -1, -6, 18, 2, -2, -3, -2, 4, 20, 2, 0, 2, 5, 7, 3, 24, 4, 2, 2, 0, -5, -12, -15, 25, 1, -3, -5, -7, -7, -2, 10, 25, 27, 2, 1, 4, 9, 16, 23, 25, 15, -10, 28, 1, -1, -2, -6, -15, -31, -54, -79, -94, -84, 32, 4, 3, 4, 6, 12, 27, 58, 112, 191, 285, 369

Offset: 0

Gus Wiseman, Oct 19 2024

Row k is the k-th differences of A013929.

			Array form:
        n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
  ---------------------------------------------------------
  k=0:   4     8     9    12    16    18    20    24    25
  k=1:   4     1     3     4     2     2     4     1     2
  k=2:  -3     2     1    -2     0     2    -3     1    -1
  k=3:   5    -1    -3     2     2    -5     4    -2     4
  k=4:  -6    -2     5     0    -7     9    -6     6    -7
  k=5:   4     7    -5    -7    16   -15    12   -13    10
  k=6:   3   -12    -2    23   -31    27   -25    23   -13
  k=7: -15    10    25   -54    58   -52    48   -36    13
  k=8:  25    15   -79   112  -110   100   -84    49     1
  k=9: -10   -94   191  -222   210  -184   133   -48   -57
Triangle form:
   4
   8   4
   9   1  -3
  12   3   2   5
  16   4   1  -1  -6
  18   2  -2  -3  -2   4
  20   2   0   2   5   7   3
  24   4   2   2   0  -5 -12 -15
  25   1  -3  -5  -7  -7  -2  10  25
  27   2   1   4   9  16  23  25  15 -10
  28   1  -1  -2  -6 -15 -31 -54 -79 -94 -84
  32   4   3   4   6  12  27  58 112 191 285 369

Initial rows: A013929, A078147, A376593.
The version for primes is A095195, noncomposites A376682, composites A377033.
A version for partitions is A175804, cf. A053445, A281425, A320590.
For squarefree numbers we have A377038, sums A377039, absolute A377040.
Triangle row-sums are A377047, absolute version A377048.
Column n = 1 is A377049, for squarefree A377041, for prime A007442 or A030016.
First position of 0 in each row is A377050.
For prime-power instead of nonsquarefree we have A377051.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
Cf. A000961, A007674, A053797, A053806, A061398, A072284, A112925, A120992, A376311, A376591, A376592.

nn=9;
t=Table[Take[Differences[NestList[NestWhile[#+1&,#+1,SquareFreeQ[#]&]&,4,2*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) A013929(i+k).

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.

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

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}]]

A378034 First-differences of A378032 (greatest number < prime(n) that is 1 or nonsquarefree).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 18 2024

nonn

Positions of 0 are A068361.
The opposite for prime-powers is A377703, differences of A345531.
For prime-powers we have A377781, differences of A065514.
The opposite is A377784, differences of A377783 (union A378040).
First-differences of A378032.
Restriction of A378036, differences of A378033.
The opposite for squarefree numbers is A378037, differences of A112926.
For squarefree numbers we have A378038, differences of A112925.
The unrestricted opposite is A378039, differences of A120327 (union A162966).
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.
A377046 encodes k-differences of nonsquarefree numbers, zeros A377050.
Cf. A053797, A053806, A072284, A065890, A224363, A377430, A377431.
Cf. A377047, A377048, A377049, A378082, A378083, A378084.

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

a(n) = A378036(prime(n)).

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)).

cp's OEIS Frontend

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

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A378033 Greatest nonsquarefree number <= n, or 1 if there is none (the case n <= 3).

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A378032 a(1) = a(2) = 1; a(n>2) is the greatest nonsquarefree number < prime(n).

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A377046 Array read by downward antidiagonals where A(n,k) is the n-th term of the k-th differences of nonsquarefree numbers.

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A377783 Least nonsquarefree number > prime(n).

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

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

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A378034 First-differences of A378032 (greatest number < prime(n) that is 1 or nonsquarefree).

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