A280892 -id:A280892 - cp's OEIS frontend

A112925 Largest squarefree integer < the n-th prime.

1, 2, 3, 6, 10, 11, 15, 17, 22, 26, 30, 35, 39, 42, 46, 51, 58, 59, 66, 70, 71, 78, 82, 87, 95, 97, 102, 106, 107, 111, 123, 130, 134, 138, 146, 149, 155, 161, 166, 170, 178, 179, 190, 191, 195, 197, 210, 222, 226, 227, 231, 238, 239, 249, 255, 262, 267, 269, 274, 278

Offset: 1

Leroy Quet, Oct 06 2005

nonn

			6 is the largest squarefree less than the 4th prime, 7. So a(4) = 6.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

For prime powers instead of squarefree numbers we have A065514, opposite A345531.
Restriction of A070321 (differences A378085) to the primes; see A378619.
The opposite is A112926, differences A378037.
Subtracting each term from prime(n) gives A240473, opposite A240474.
For nonsquarefree numbers we have A378033, differences A378036, see A378034, A378032.
For perfect powers we have A378035.
First differences are A378038.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259.
A013928 counts squarefree numbers up to n - 1.
A013929 lists the nonsquarefree numbers, differences A078147.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
A112929 counts squarefree numbers up to prime(n).
Cf. A061400, A112928, A112930.
Cf. A007674, A007947, A045882, A053797, A053806, A067535, A072284, A073247, A081217, A175216, A280892, A377783.

with(numtheory): a:=proc(n) local p,B,j: p:=ithprime(n): B:={}: for j from 1 to p-1 do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: B[nops(B)] end: seq(a(m),m=1..75); # Emeric Deutsch, Oct 14 2005

With[{k = 120}, Table[SelectFirst[Range[Prime@ n - 1, Prime@ n - Min[Prime@ n - 1, k], -1], SquareFreeQ], {n, 60}]] (* Michael De Vlieger, Aug 16 2017 *)

a(n,p=prime(n))=while(!issquarefree(p--),); p \\ Charles R Greathouse IV, Aug 16 2017

a(n) = prime(n) - A240473(n). - Gus Wiseman, Jan 10 2025

More terms from Emeric Deutsch, Oct 14 2005

A112926 Smallest squarefree integer > the n-th prime.

Original entry on oeis.org

3, 5, 6, 10, 13, 14, 19, 21, 26, 30, 33, 38, 42, 46, 51, 55, 61, 62, 69, 73, 74, 82, 85, 91, 101, 102, 105, 109, 110, 114, 129, 133, 138, 141, 151, 154, 158, 165, 170, 174, 181, 182, 193, 194, 199, 201, 213, 226, 229, 230, 235, 241, 246, 253, 258, 265, 271, 273

Offset: 1

Leroy Quet, Oct 06 2005

nonn

			10 is the smallest squarefree number greater than the 4th prime, 7. So a(4) = 10.
From _Gus Wiseman_, Dec 07 2024: (Start)
The first number line below shows the squarefree numbers. The second shows the primes:
--1--2--3-----5--6--7-------10-11----13-14-15----17----19----21-22-23-------26--
=====2==3=====5=====7==========11====13==========17====19==========23===========
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A061400, A112928, A112929, A112930.
Restriction of A067535, differences A378087.
The unrestricted opposite is A070321, differences A378085.
The opposite is A112925, differences A378038.
Subtracting prime(n) from each term gives A240474, opposite A240473.
For nonsquarefree we have A377783, restriction of A120327.
The nonsquarefree differences are A377784, restriction of A378039.
First differences are A378037.
For perfect power we have A378249, A378617, A378250, A378251.
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, A053797, A053806, A072284, A073247, A175216, A280892, A345531, A378032, A378618.

with(numtheory): a:=proc(n) local p,B,j: p:=ithprime(n): B:={}: for j from p+1 to p+20 do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: B[1] end: seq(a(m),m=1..75); # Emeric Deutsch, Oct 10 2005

Do[k = Prime[n] + 1; While[ !SquareFreeQ[k], k++ ]; Print[k], {n, 1, 100}] (* Ryan Propper, Oct 10 2005 *)
With[{k = 120}, Table[SelectFirst[Range[Prime@ n + 1, Prime@ n + k], SquareFreeQ], {n, 58}]] (* Michael De Vlieger, Aug 16 2017 *)

a(n,p=prime(n))=while(!issquarefree(p++),); p \\ Charles R Greathouse IV, Aug 16 2017

a(n) = prime(n) + A240474(n). - Gus Wiseman, Dec 07 2024

More terms from Ryan Propper and Emeric Deutsch, Oct 10 2005

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}]]

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}]]

A067874 Positive integers x satisfying x^2 - D*y^2 = 1 for a unique integer D.

Original entry on oeis.org

2, 4, 6, 12, 14, 16, 18, 20, 22, 30, 32, 34, 36, 38, 40, 42, 52, 54, 56, 58, 60, 66, 68, 70, 72, 78, 84, 86, 88, 90, 92, 94, 96, 102, 104, 106, 108, 110, 112, 114, 128, 130, 132, 138, 140, 142, 144, 150, 156, 158, 160, 162, 164, 166, 178, 180, 182, 184, 186, 192, 194, 196, 198

Offset: 1

Lekraj Beedassy, Feb 25 2002

D is unique iff x^2 - 1 is squarefree, in which case it follows with necessity that D=x^2-1 and y=1.
All terms are even. A014574 is a subsequence.
Conjecture: All terms of A002110 > 1 are a subsequence. - Griffin N. Macris, Apr 11 2016
All n such that n+1 and n-1 are in A056911. - Robert Israel, Apr 12 2016
The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - Amiram Eldar, Feb 25 2021

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A002110, A005117, A014574, A056911, A065474, A226993, A272799, A280892, A379971 (characteristic function).

Subsequence of A379965.

[n: n in [1..110] | IsSquarefree(n-1) and IsSquarefree(n+1)]; // Juri-Stepan Gerasimov, Jan 17 2017

select(t -> numtheory:-issqrfree(t^2-1), [seq(n,n=2..1000,2)]); # Robert Israel, Apr 12 2016

Select[Range[200], SquareFreeQ[#^2-1]&] (* Vladimir Joseph Stephan Orlovsky, Oct 26 2009 *)

from itertools import count, islice
from sympy import factorint
def A067874_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda k:max(factorint(k-1).values(),default=1)==1 and max(factorint(k+1).values())==1, count(max(startvalue+(startvalue&1),2),2))
A067874_list = list(islice(A067874_gen(),20)) # Chai Wah Wu, Apr 24 2024

a(n) = 2*A272799(n). - Juri-Stepan Gerasimov, Jan 17 2017

Corrected and extended by Max Alekseyev, Apr 26 2009

Further edited by Max Alekseyev, Apr 28 2009

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}]]

A281192 Numbers with no squarefree neighbors.

Original entry on oeis.org

17, 19, 26, 49, 51, 53, 55, 89, 91, 97, 99, 125, 127, 149, 151, 161, 163, 170, 197, 199, 233, 235, 241, 243, 244, 249, 251, 269, 271, 293, 295, 305, 307, 337, 339, 341, 343, 349, 351, 362, 377, 379, 413, 415, 424, 449, 451, 476, 485, 487, 489, 491, 521, 523, 530, 549, 551, 557, 559, 577

Offset: 1

Juri-Stepan Gerasimov, Jan 16 2017

Includes all k == 17 or 19 (mod 36), also 2*p^2-1 and 2*p^2+1 for odd primes p. - Robert Israel, Jan 17 2017
This sequence has density around 0.106.... - Charles R Greathouse IV, Jan 23 2017
More accurately, the asymptotic density of this sequence is 1 - 2/zeta(2) + Product_{p prime} (1 - 2/p^2) = 1 - 2 * A059956 + A065474 = 0.1067798952... - Amiram Eldar, Feb 25 2021

			17 is in this sequence because 17 - 1 = 16 = 2^4 and 17 + 1 = 18 = 2*3^2 are not squarefree.

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

Supersequence of A075432 and A235578.

Cf. A005117, A059956, A065474, A280892.

[n: n in [2..600] | not IsSquarefree(n-1) and not IsSquarefree(n+1)];

select(t -> not numtheory:-issqrfree(t-1) and not numtheory:-issqrfree(t+1), [$1..1000]); # Robert Israel, Jan 17 2017

Select[Range[600], !SquareFreeQ[# - 1] && !SquareFreeQ[# + 1] &] (* Vincenzo Librandi, Jan 17 2017 *)

is(n)=!issquarefree(n-1) && !issquarefree(n+1) \\ Charles R Greathouse IV, Jan 23 2017

A378619 Distance between n and the greatest squarefree number <= n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 12 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..30000

Positions of 0 are A005117.
Positions of first appearances are A020755 - 1.
Positions of 1 are A053806.
Subtracting each term from n gives A070321.
The opposite version is A081221.
Restriction to the primes is A240473, opposite A240474.
A013929 lists the nonsquarefree numbers, differences A078147.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
Cf. A007674, A013928, A053797, A072284, A073247, A076259, A280892.
Cf. A007947, A067535, A081217, A112925.

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

A378619(n) = forstep(k=n,1,-1,if(issquarefree(k), return(n-k))); \\ Antti Karttunen, Jan 29 2025

from itertools import count
from sympy import factorint
def A378619(n): return n-next(m for m in count(n,-1) if max(factorint(m).values(),default=0)<=1) # Chai Wah Wu, Dec 14 2024

a(n) = n - A070321(n).

Data section extended to a(105) by Antti Karttunen, Jan 29 2025

A349230 Numbers k such that k and k+3 are consecutive squarefree numbers.

Original entry on oeis.org

7, 23, 26, 43, 62, 74, 79, 115, 119, 134, 146, 151, 167, 170, 174, 187, 206, 223, 259, 274, 278, 287, 295, 314, 323, 331, 359, 362, 367, 374, 386, 403, 439, 458, 494, 506, 511, 523, 527, 530, 538, 566, 574, 583, 619, 623, 635, 638, 655, 674, 691, 710, 727, 734

Offset: 1

Amiram Eldar, Nov 11 2021

nonn

The asymptotic density of this sequence is 0.0716601371276261... (Mossinghoff et al., 2021).

			7 is a term since 7 and 7 + 3 = 10 = 2*5 are squarefree, and 7 + 1 = 8 = 2^3 and 7 + 2 = 9 = 3^2 are not.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Michael J. Mossinghoff, Tomás Oliveira e Silva, and Tim Trudgian, The distribution of k-free numbers, Mathematics of Computation, Vol. 90, No. 328 (2021), pp. 907-929; arXiv preprint, arXiv:1912.04972 [math.NT], 2019-2020.

Cf. A005117, A007674, A020754, A280892, A349231.

Select[Range[750], Boole[SquareFreeQ /@ (# + {0, 1, 2, 3})] == {1, 0, 0, 1} &]

isok(k) = issquarefree(k) && !issquarefree(k+1) && !issquarefree(k+2) && issquarefree(k+3); \\ Michel Marcus, Nov 11 2021

cp's OEIS Frontend

A112925 Largest squarefree integer < the n-th prime.

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A112926 Smallest squarefree integer > the n-th prime.

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

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A067874 Positive integers x satisfying x^2 - D*y^2 = 1 for a unique integer D.

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

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A281192 Numbers with no squarefree neighbors.

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