A061399 -id:A061399 - cp's OEIS frontend

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

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)).

A373409 Length of the n-th maximal antirun of nonsquarefree numbers differing by more than one.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 06 2024

nonn

An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.

Conjecture: The maximum is 9, and there is no antirun of more than 9 nonsquarefree numbers. Confirmed up to 100,000,000.

			Row-lengths of:
   4   8
   9  12  16  18  20  24
  25  27
  28  32  36  40  44
  45  48
  49
  50  52  54  56  60  63
  64  68  72  75
  76  80
  81  84  88  90  92  96  98
  99
The first maximal antirun of length 9 is the following, shown with prime indices:
  6345: {2,2,2,3,15}
  6348: {1,1,2,9,9}
  6350: {1,3,3,31}
  6352: {1,1,1,1,78}
  6354: {1,2,2,71}
  6356: {1,1,4,49}
  6358: {1,5,7,7}
  6360: {1,1,1,2,3,16}
  6363: {2,2,4,26}

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

Positions of first appearances are A373573, sorted A373574.
Functional neighbors: A027833, A053797, A068781, A373127, A373403, A373410, A373412.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A025157, A049093, A061398, A061399, A077641, A077643, A110969, A174965, A294242, A350842, A373198.

Length/@Split[Select[Range[1000],!SquareFreeQ[#]&],#1+1!=#2&]//Most

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 *)

A375702 Length of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

Original entry on oeis.org

2, 3, 6, 8, 1, 4, 3, 12, 14, 16, 18, 20, 3, 2, 15, 24, 26, 19, 8, 17, 12, 32, 34, 18, 17, 38, 40, 42, 27, 16, 46, 48, 50, 52, 54, 56, 58, 60, 38, 23, 64, 66, 68, 70, 34, 37, 74, 76, 78, 80, 46, 35, 84, 86, 88, 22, 67, 70, 9, 11, 94, 96, 98, 100, 102, 39, 64

Offset: 1

Gus Wiseman, Aug 27 2024

nonn

Non-perfect-powers A007916 are numbers with no proper integer roots.

			The list of all non-perfect-powers, split into runs, begins:
   2   3
   5   6   7
  10  11  12  13  14  15
  17  18  19  20  21  22  23  24
  26
  28  29  30  31
  33  34  35
  37  38  39  40  41  42  43  44  45  46  47  48
Row n has length a(n), first A375703, last A375704, sum A375705.

For nonsquarefree numbers we have A053797, anti-runs A373409.
For squarefree numbers we have A120992, anti-runs A373127.
For nonprime numbers we have A176246, anti-runs A373403.
For prime-powers we have A373675, anti-runs A373576.
For non-prime-powers we have A373678, anti-runs A373679.
The anti-run version is A375736, sum A375737.
For runs of non-perfect-powers (A007916):
- length: A375702 (this).
- first: A375703
- last: A375704
- sum: A375705
A001597 lists perfect-powers, differences A053289.
A007916 lists non-perfect-powers, differences A375706.
A046933 counts composite numbers between primes.
Cf. A007674, A045542, A061399, A216765, A251092, A375708, A375714.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Length/@Split[Select[Range[100],radQ],#1+1==#2&]//Most

For n > 2 we have a(n) = A053289(n+1) - 1.

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.

A375704 Maximum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

Original entry on oeis.org

3, 7, 15, 24, 26, 31, 35, 48, 63, 80, 99, 120, 124, 127, 143, 168, 195, 215, 224, 242, 255, 288, 323, 342, 360, 399, 440, 483, 511, 528, 575, 624, 675, 728, 783, 840, 899, 960, 999, 1023, 1088, 1155, 1224, 1295, 1330, 1368, 1443, 1520, 1599, 1680, 1727, 1763

Offset: 1

Gus Wiseman, Aug 29 2024

nonn

Non-perfect-powers (A007916) are numbers with no proper integer roots.

Also numbers k > 0 such that k is a perfect power (A001597) but k+1 is not.

			The list of all non-perfect-powers, split into runs, begins:
   2   3
   5   6   7
  10  11  12  13  14  15
  17  18  19  20  21  22  23  24
  26
  28  29  30  31
  33  34  35
  37  38  39  40  41  42  43  44  45  46  47  48
Row n begins with A375703(n), ends with a(n), adds up to A375705(n), and has length A375702(n).

For nonprime numbers: A006093, min A055670, anti-runs A068780, min A005381.
For prime numbers we have A045344.
Inserting 8 after 7 gives A045542.
For nonsquarefree numbers we have A072284(n) + 1, anti-runs A068781.
For squarefree numbers we have A373415, anti-runs A007674.
For prime-powers we have A373674 (min A373673), anti-runs A006549 (A120430).
Non-prime-powers: A373677 (min A373676), anti-runs A255346 (min A373575).
The anti-run version is A375739.
A001597 lists perfect-powers, differences A053289.
A046933 counts composite numbers between primes.
A375736 gives lengths of anti-runs of non-prime-powers, sums A375737.
For runs of non-perfect-powers (A007916):
- length: A375702 = A053289(n+1) - 1
- first: A375703 (same as A216765 with 2 exceptions)
- last: A375704 (this) (same as A045542 with 8 removed)
- sum: A375705
Cf. A053797, A053806, A061398, A061399, A251092, A373408, A375708, A375714.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Max/@Split[Select[Range[100],radQ],#1+1==#2&]//Most
- or -
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Select[Range[100],radQ[#]&&!radQ[#+1]&]

For n > 2 we have a(n) = A045542(n+1).

A373412 Sum of the n-th maximal antirun of nonsquarefree numbers differing by more than one.

Original entry on oeis.org

12, 99, 52, 180, 93, 49, 335, 279, 156, 629, 99, 540, 237, 245, 125, 521, 567, 450, 963, 340, 347, 728, 1386, 1080, 1637, 243, 244, 1511, 1610, 555, 852, 1171, 2142, 960, 985, 1689, 343, 1042, 351, 1068, 724, 732, 1116, 1905, 1980, 2898, 424, 2161, 3150, 2339

Offset: 1

Gus Wiseman, Jun 06 2024

nonn

The length of this antirun is given by A373409.

An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.

			Row-sums of:
   4   8
   9  12  16  18  20  24
  25  27
  28  32  36  40  44
  45  48
  49
  50  52  54  56  60  63
  64  68  72  75
  76  80
  81  84  88  90  92  96  98
  99

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

The partial sums are a subset of A329472.
Functional neighbors: A068781, A373404, A373405, A373409, A373410, A373411, A373414.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A025157, A049093, A049094, A061399, A101836, A112925, A143658, A294242, A350842, A372683, A373197.

Total/@Split[Select[Range[100],!SquareFreeQ[#]&],#1+1!=#2&]//Most

cp's OEIS Frontend

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

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A378086 Number of nonsquarefree numbers < prime(n).

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A373409 Length of the n-th maximal antirun of nonsquarefree numbers differing by more than one.

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

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A375702 Length of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

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

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A375704 Maximum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

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