A013929 -id:A013929 - cp's OEIS frontend

A375927 Numbers k such that A005117(k+1) - A005117(k) = 1. In other words, the k-th squarefree number is 1 less than the next.

1, 2, 4, 5, 7, 9, 10, 14, 15, 18, 19, 21, 22, 24, 25, 27, 28, 30, 35, 36, 38, 40, 41, 43, 44, 46, 48, 49, 51, 53, 54, 58, 59, 62, 63, 65, 66, 68, 69, 71, 72, 74, 76, 79, 80, 82, 84, 85, 87, 88, 90, 94, 96, 97, 101, 102, 105, 107, 108, 110, 111, 113, 114, 116

Offset: 1

Gus Wiseman, Sep 12 2024

nonn

The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^2-1)) = 0.53071182... (A065469). - Amiram Eldar, Sep 15 2024

			The squarefree numbers are 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ... which first increase by one after terms 1, 2, 4, 5, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Positions of 1's in A076259.
For prime-powers (A246655) we have A375734.
First differences are A373127.
For nonsquarefree instead of squarefree we have A375709.
For nonprime numbers we have A375926, differences A373403.
For composite numbers we have A375929.
The complement is A375930, differences A120992.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A061399, A065469, A072284, A110969, A294242, A373409, A373410, A373573.

Join@@Position[Differences[Select[Range[100],SquareFreeQ[#]&]],1]

lista(kmax) = {my(is1 = 1, is2, c = 1); for(k = 2, kmax, is2 = issquarefree(k); if(is2, c++); if(is1 && is2, print1(c-1, ", ")); is1 = is2);} \\ Amiram Eldar, Sep 15 2024

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}]]

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}]]

A373199 Least k such that the k-th maximal run of nonsquarefree numbers has length n. Position of first appearance of n in A053797.

Original entry on oeis.org

1, 2, 13, 68, 241, 6278, 61921, 311759, 2530539

Offset: 1

Gus Wiseman, Jun 08 2024

A run of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by one. The a(n)-th run of nonsquarefree numbers begins with A045882 = A051681, subset of A053806.

			The maximal runs of nonsquarefree numbers begin:
   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
The a(n)-th rows are:
     4
     8     9
    48    49    50
   242   243   244   245
   844   845   846   847   848
For example, (48, 49, 50) is the first maximal run of 3 nonsquarefree numbers, so a(3) = 13.

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

For composite instead of nonsquarefree we have A073051.
The version for squarefree runs is A373128.
For prime instead of nonsquarefree we have A373400.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A007674, A020754, A045882, A120992, A061398, A061399, A068781, A101836, A251092, A294242, A373410, A373412.

seq=Length/@Split[Select[Range[10000],!SquareFreeQ[#]&],#1+1==#2&];
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[y,Range[#]]&];
Table[Position[seq,i][[1,1]],{i,spna[seq]}]

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

A375927 Numbers k such that A005117(k+1) - A005117(k) = 1. In other words, the k-th squarefree number is 1 less than the next.

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

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

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

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A373199 Least k such that the k-th maximal run of nonsquarefree numbers has length n. Position of first appearance of n in A053797.

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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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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).