A045882 -id:A045882 - cp's OEIS frontend

A020754 Increasing gaps between squarefree numbers (lower end).

1, 3, 7, 47, 241, 843, 22019, 217069, 1092746, 8870023, 221167421, 47255689914, 82462576219, 1043460553363, 79180770078547, 3215226335143217, 23742453640900971, 125781000834058567

Offset: 1

David W. Wilson

We only consider gaps that set new records. The first gap of size 12 occurs (at 221167421) before the first gap of size 11 (at 262315466) and so for n>10, the n-th term in this sequence does not correspond to the first gap of length n. See A020753. - Nathan McNew, Dec 02 2020

The length of these runs are significantly shorter than would be predicted by a naive random model (for such a model see, e.g., Gordon, Schilling, & Waterman). For example, with n = a(18) and p = 6/Pi^2 the expected largest run is about 77.9 with variance 6.7, while A020753(18) = 18 which is 23 standard deviations smaller. - Charles R Greathouse IV, Oct 29 2021

			The first gap in A005117 occurs between 1 and 2 and has length 1. The next largest gap occurs between 3 and 5 and has length 2. The next largest gap is between 7 and 10 and has length 3. Etc.

Tsz Ho Chan, New small gaps between squarefree numbers, arXiv:2110.09990 [math.NT], 2021. [Note: according to Pandey, Chan has discovered an error in this paper.]
Louis Gordon, Mark F. Schilling, and Michael S. Waterman, An extreme value theory for long head runs, Probability Theory and Related Fields, Vol. 72 (1986), pp. 279-287.
Angel Kumchev, Wade McCormick, Nathan McNew, Ariana Park, Russell Scherr, and Simon Ziehr, Explicit bounds for large gaps between squarefree and cubefree integers, arXiv preprint (2022). arXiv:2211.09975 [math.NT]
Michael J. Mossinghoff, Tomás Oliveira e Silva, and Tim Trudgian, The distribution of k-free numbers, arXiv:1912.04972 [math.NT], 2019. See Table 3, p. 14.
Mayank Pandey, Squarefree numbers in short intervals, arXiv preprint (2024). arXiv:2401.13981 [math.NT]

Cf. A005117, A020753, A020755, A045882, A051681.

Module[{nn=11*10^5,sf,df},sf=Select[Range[nn],SquareFreeQ];df=Differences[sf];DeleteDuplicates[ Thread[{Most[sf],df}],GreaterEqual[#1[[2]],#2[[2]]]&]][[;;,1]] (* Harvey P. Dale, May 24 2023 *)

A020754(n)=for(k=L=1, 9e9, issquarefree(k)||next; k-L>=n&&return(L); L=k) \\ For illustrative purpose only, not useful for n>10. - M. F. Hasler, Dec 28 2015

r=0; L=1; forsquarefree(n=2,10^8,t=n[1]-L; if(t>r,r=t; print1(L", ")); L=n[1]) \\ Charles R Greathouse IV, Oct 22 2021

a(n) = A020755(n) - A020753(n); also a(n) = A020754(n+[n>10]) - 1 at least for n < 19. - M. F. Hasler, Dec 28 2015

Thanks to Christian G. Bower for additional comments.

a(16)-a(18) from A045882 by Jens Kruse Andersen, May 01 2015

A051681 Smallest term of first run of exactly n consecutive integers which are not squarefree.

Original entry on oeis.org

4, 8, 48, 242, 844, 22020, 217070, 1092747, 8870024, 262315467, 221167422, 47255689915, 82462576220, 1043460553364, 79180770078548, 3215226335143218, 23742453640900972, 125781000834058568

Offset: 1

Louis Marmet (louis(AT)marmet.org) and David Bernier (ezcos(AT)yahoo.com)

			a(5) = 844: 844 = 2^2*211, 845 = 5*13^2, 846 = 2*3^2*47, 847 = 7*11^2, 848 = 2^4*53.

a(16) was obtained as a result of a team effort by Z. McGregor-Dorsey et al.

L. Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012. See also the author page.
"sikefield3", double-square (2019).
Eric Weisstein's World of Mathematics, Squarefree.

Cf. A045882 (another version), A013929, A053806.

Module[{tb=Table[If[SquareFreeQ[n],0,1],{n,11*10^5}]},Table[ SequencePosition[ tb,PadRight[{},k,1],1][[All,1]],{k,8}]]//Flatten (* The program generates the first 8 terms of the sequence. To generate more, increase the constants for n and k but the program may take a long time to run. *) (* Harvey P. Dale, Mar 24 2022 *)

iscons(x, n)=if (issquarefree(x-1) && issquarefree(x+n), for (k = 0, n-1, if (issquarefree(x+k), return (0));); return (1);); return (0);
a(n) = {my(x = 1); while (! iscons(x, n), x++); x;} \\ Michel Marcus, Jan 13 2014

a(16) reported by Louis Marmet (louis(AT)marmet.org), Jul 24 2000
a(17) was obtained as a result of a team effort by E. Wong et al.
a(18) = 125781000834058568 was obtained as a result of a team effort by L. Marmet et al.

A081221 Number of consecutive numbers >= n having at least one square divisor > 1.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 10 2003

nonn

The first time terms 0..7 occur is at n = 1, 4, 8, 48, 242, 844, 22020, 217070. - Antti Karttunen, Sep 22 2017

			For n = 3, 3 is a squarefree number, thus a(3) = 0.
For n = 48, neither 48 = 2^4 * 3 nor 49 = 7^2, nor 50 = 2^2 * 5 are squarefree, but 51 = 3*17 is, thus a(48) = 3. - _Antti Karttunen_, Sep 22 2017

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A005117, A008683, A045882, A051681, A068781, A067535, A080733.

Flatten@ Map[If[First@ # == 0, #, Reverse@ Range@ Length@ #] &, SplitBy[Table[DivisorSum[n, 1 &, And[# > 1, IntegerQ@ Sqrt@ #] &], {n, 120}], # > 0 &]] (* Michael De Vlieger, Sep 22 2017 *)

A081221(n) = { my(k=0); while(!issquarefree(n+k),k++); k; }; \\ Antti Karttunen, Sep 22 2017

from itertools import count
from sympy import factorint
def A081221(n): return next(m for m in count(0) if max(factorint(n+m).values(),default=0)<=1) # Chai Wah Wu, Dec 04 2024

mu(k) = 0 for n <= k < n+a(n) and mu(n+a(n)) <> 0, where mu = A008683 (Moebius function).
a(n)*mu(n) = 0.
a(A068781(n)) > 0.
a(n) = A067535(n) - n. - Amiram Eldar, Oct 10 2023

A271443 Earliest start of a run of n numbers divisible by a cube larger than one.

Original entry on oeis.org

8, 80, 1375, 22624, 18035622, 4379776620, 1204244328624, 2604639091138248, 2604639091138248

Offset: 1

Giovanni Resta, Apr 23 2016

a(5)-a(7) were found by Donovan Johnson.

			a(9) = 2604639091138248 and the following 8 numbers are divisible by 2^3, 11^3, 5^3, 17^3, 7^3, 13^3, 3^3, 19^3, and 2^4, respectively.

Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., (2009), page 63.

Cf. A046099, A068140, A122692, A271444, A271445, A271446, A271447.

Cf. A045882, A330480, A330481, A330482, A330486, A330483, A330484, A330485.

a[n_] := Block[{k=1, c=0}, While[ c

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}]]

A070284 Smallest of 4 consecutive numbers each divisible by a square.

Original entry on oeis.org

242, 844, 845, 1680, 1681, 2888, 2889, 3174, 3624, 3625, 3750, 5046, 5047, 8475, 8523, 8954, 10050, 10827, 10924, 10925, 11322, 13374, 14748, 14749, 15775, 15848, 15849, 16575, 17404, 17405, 19647, 19940, 19941, 20574, 21462

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 09 2002

nonn

This sequence has positive density in N; the density is around 0.0025.

The sequence includes an infinite family of arithmetic progressions. Such AP's can be constructed to each term, with large differences [like e.g. square of primorials, A061742]. It is necessary to solve suitable systems of linear Diophantine equations. E.g.: subsequences of quadruples of terms = {44100k+29349, 44100k+29350, 44100k+29351, 44100k+29352} = {9(4900k+3261), 25(1764k+1174), 49(900k+599), 4(11025k+7338)}; starting terms in this sequence = {29349, 73449, 117549, ...}; difference = A002110(4)^2 = 210^2. - Labos Elemer, Nov 25 2002

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A070258, A068781, A045882, A049535, A077647, A078143.

f[n_] := Union[Transpose[FactorInteger[n]][[2]]][[ -1]]; a = 0; b = 1; c = 0; Do[d = f[n]; If[a > 1 && b > 1 && c > 1 && d > 1, Print[n - 3]]; a = b; b = c; c = d, {n, 4, 10^6}]
Flatten[Position[Partition[SquareFreeQ/@Range[60000],4,1],?(Union[#] == {False}&),{1},Heads->False]] (* _Harvey P. Dale, May 24 2014 *)

is(n)=for(i=n,n+3, if(!issquarefree(n), return(0))); 1 \\ Charles R Greathouse IV, Sep 14 2015

A070284 = { A070258[k] | A070258[k+1] = A070258[k]+1 }. - M. F. Hasler, Feb 01 2016

More terms from Robert G. Wilson v, May 09 2002

b-file from Charles R Greathouse IV, Jul 23 2010

A049535 Starts of runs of exactly 6 consecutive nonsquarefree numbers.

Original entry on oeis.org

22020, 24647, 30923, 47672, 55447, 57120, 73447, 74848, 96675, 105772, 121667, 121847, 152339, 171348, 179972, 182347, 185247, 190447, 200848, 204323, 215303, 229172, 233223, 234375, 240424, 268223, 274547, 310120, 327424, 338920

Offset: 1

Labos Elemer

nonn

			Squares dividing the numbers in the starting at 22020 are 4, 361, 121, 9, 4, 25, respectively.

Robert Israel, Table of n, a(n) for n = 1..1000

The smallest members of such strings of length k are listed in A045882.
Cf. A001694 powerful numbers, A013929 not squarefree.
Cf. A045882 (min terms), A068781 (2-chains), A070258 (3-chains), A070284 (4-chains), A078144 (5-chains), A049535 (6-chains), A077640 (7-chains), A077647 (8-chains), A078143 (9-chains), A268313 (10-chains), A268314 (11-chains).

Res:= NULL:
st:= 0:
for n from 1 to 500000 do
  if numtheory:-issqrfree(n) then
    if st = 6 then Res:= Res, n-6 fi;
    st:= 0;
  else
    st:= st+1;
  fi
od:
Res; # Robert Israel, Feb 08 2017

Select[Range[400000], !SquareFreeQ[#] && !SquareFreeQ[#+1] && !SquareFreeQ[#+2] && !SquareFreeQ[#+3] && !SquareFreeQ[#+4] && !SquareFreeQ[#+5] && SquareFreeQ[#+6]&] (* Vladimir Joseph Stephan Orlovsky, Mar 30 2011 *)
Flatten[Position[Partition[SquareFreeQ/@Range[60000],6,1],?(Union[#] == {False}&),{1},Heads->False]] (* _Harvey P. Dale, May 24 2014 *)

{ A078144(k) | A078144(k+1) = A078144(k)+1 and A078144(k+2) > A078144(k)+2 }. - M. F. Hasler, Feb 01 2016

Definition corrected by Donald S. McDonald, Nov 07 2002

Corrected by Robert Israel, Feb 08 2017

A077647 Smallest term of a run of at least 8 consecutive integers which are not squarefree.

Original entry on oeis.org

1092747, 7216618, 8870024, 8870025, 14379271, 22635347, 24816974, 25047846, 33678771, 33908368, 33908369, 34394371, 34682346, 37923938, 49250144, 49250145, 53379270, 69147868, 69147869, 70918820, 70918821, 71927247, 72913022, 83605071, 85972019, 90571646

Offset: 1

Labos Elemer, Nov 18 2002

nonn

			n=8870024: squares dividing n+j (j=0...8) i.e. 9 consecutive integers are as follows {4,25,121,841,4,49,961,9,16}

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

Cf. A013929, A051681.

Cf. A045882 (first k-chain), A068781 (2-chains), A070258 (3-chains), A070284 (4-chains), A078144 (5-chains), A049535 (6-chains), A077640 (7-chains), A077647 (8-chains), A078143 (9-chains), A268313 (10-chains), A268314 (11-chains).

s8[x_] := Apply[Plus, Table[Abs[MoebiusMu[x+j]], {j, 0, 7}]]; Do[If[s8[n] == 0, Print[n]], {n, 10^8}]
Flatten[Position[Partition[SquareFreeQ/@Range[91000000],8,1],_?(Union[#]=={False}&),{1},Heads->False]]

for(n=1,10^8,forstep(k=7,0,-1,issquarefree(n+k)&&(n+=k)&&next(2));print1(n",")) \\ M. F. Hasler, Feb 03 2016

A077647 = { A077640[k] | A077640[k+1] = A077640[k]+1 }. - M. F. Hasler, Feb 01 2016

A078143 Smallest term of a run of at least 9 consecutive integers which are not squarefree.

Original entry on oeis.org

8870024, 33908368, 49250144, 69147868, 70918820, 111500620, 112931372, 164786748, 167854344, 200997948, 203356712, 207543320, 211014920, 216785256, 221167422, 221167423, 221167424, 236645624, 240574368, 262315467, 262315468

Offset: 1

Labos Elemer, Nov 22 2002

nonn

The sequence includes an infinite family of arithmetic progressions. Such AP's can be constructed to each term, with large differences [like squares of primorials, A061742(7)]. It is necessary to solve suitable systems of linear Diophantine equations. E.g.: arithmetic progression subsequences of starting 9-chains is {mk+69147868+j} where j=0..8, m=510510^2 because square prime factors of a(4)+j=68147868+j are 4, 49, 121, 169, 4, 9, 289, 25, 4 resp. for j=0..8; k goes to infinity; 7th primorial is sufficient, 9th is not necessary. Construction is provable for arbitrary long [>9] chains. - Labos Elemer, Nov 25 2002

More precisely, if in one run {a(n)+j, j=0..8} the maximum smallest square factor is p^2, then an infinite subsequence is given by {a(n)+(p#)^2*k, k=0..oo}, where p# = A034386(p). One may get a smaller step taking the least L^2 which has a square factor in common with each of the 9 consecutive terms. - M. F. Hasler, Feb 03 2016

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A013929, A045882 (first of the k-chains), A051681.

Cf. A068781 (2-chains), A070258 (3-chains), A070284 (4-chains), A078144 (5-chains), A049535 (6-chains), A077640 (7-chains), A077647 (8-chains), A078143 (9-chains), A268313 (10-chains), A268314 (11-chains).

s9[x_] := Apply[Plus, Table[Abs[MoebiusMu[x+j]], {j, 0, 8}]]; Do[If[Equal[s9[n], 0], Print[n]], {n, 8000000, 1000000000}]

is(n)=for(i=n,n+8, if(!issquarefree(i), return(0))); 1 \\ Charles R Greathouse IV, Nov 05 2017

A078143 = { A077647[k] | A077647[k+1] = A077647[k]+1 } = { A077640[k] | A077640[k+2] = A077640[k]+2 } = { A078144[k] | A078144[k+4] = A078144[k]+4 } etc. Note that A049535 is defined differently. - M. F. Hasler, Feb 01 2016

a(n) < 4666864390*n. With more work this bound can be decreased significantly. - Charles R Greathouse IV, Nov 05 2017

a(6)-a(21) from Donovan Johnson, Nov 26 2008

A078144 Starts for strings of at least five consecutive nonsquarefree numbers.

Original entry on oeis.org

844, 1680, 2888, 3624, 5046, 10924, 14748, 15848, 17404, 19940, 22020, 22021, 22624, 23272, 24647, 24648, 25772, 29348, 30248, 30923, 30924, 33172, 36700, 37248, 38724, 39444, 40472, 45372, 47672, 47673, 47724, 47824, 48372, 49488

Offset: 1

Labos Elemer, Nov 25 2002

nonn

			Squares dividing 5-string=844+j, j=0,..,4 are as follows:4,169,9,121,16 resp. Each term initiates an arithmetic progression with suitable large difference. Such progressions are constructible by solving suitable linear Diophantine equations. E.g., quintet = {m*k+3689649, m*k+3689650, m*k+3689651, m*k+3689652, m*k+3689653} = {9*(592900*k+409961), 25*(213444*k+147586), 49*(108900*k+75299), 4*(1334025*k+922413), 121*(44100*k+30493)}; m=2310*2310=A002110(5)^2=A061742(5)=5336100.

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

Cf. A045882 (min terms), A068781 (2-chains), A070258 (3-chains), A070284 (4-chains), A078144 (5-chains), A049535 (6-chains), A077647 (8-chains), A078143 (9-chains), A188296.

s5[x_] := Total[Table[Abs[MoebiusMu[x + j]], {j, 0, 4}]] == 0; Select[Range[50000], s5]
Flatten[Position[Partition[SquareFreeQ/@Range[60000],5,1],?(Union[#] == {False}&),{1},Heads->False]] (* _Harvey P. Dale, May 24 2014 *)
SequencePosition[Table[If[SquareFreeQ[n],0,1],{n,50000}],{1,1,1,1,1}][[All,1]] (* Harvey P. Dale, Oct 16 2022 *)

Equals { A070284[k] | A070284[k+1] = A070284[k]+1 }. - M. F. Hasler, Feb 01 2016

a(n) = A188296(n) - 2. - Amiram Eldar, Feb 09 2021

cp's OEIS Frontend

A020754 Increasing gaps between squarefree numbers (lower end).

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A051681 Smallest term of first run of exactly n consecutive integers which are not squarefree.

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A081221 Number of consecutive numbers >= n having at least one square divisor > 1.

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A271443 Earliest start of a run of n numbers divisible by a cube larger than one.

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

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A070284 Smallest of 4 consecutive numbers each divisible by a square.

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A049535 Starts of runs of exactly 6 consecutive nonsquarefree numbers.

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