A051681 -id:A051681 - cp's OEIS frontend

A055556 Differences of arithmetic progressions of which the terms give chains of n consecutive nonsquarefree numbers if started with terms of A045882(n) or A051681(n).

4, 72, 58800, 1757711340, 54169994838960, 950645196424756293060600, 94630125612279498512066506747436400, 14119054639791549212725337736060964803166626000

Offset: 1

Labos Elemer, Jul 10 2000

nonn

			For n=5, the difference of the relevant progression is 54169994838960.

M. Filaseta and O. Trifonov, On Gaps between Squarefree Numbers. In Analytic Number Theory, Vol 85, 1990, Birkhauser, Basel, pp. 235-253.
E. Fogels, On the average values of arithmetic functions, Proc. Cambridge Philos. Soc. 1941, 37: 358-372.
K. F. Roth, On the gaps between squarefree numbers, J. London Math. Soc. 1951 (2) 26:263-268.

Cf. A013929, A051681, A045882, A005117, A053797, A005117, A053806.

a(n) = LCM(x, x+1, ..., x+n-1), where x = A045882(n).

A053806 Numbers where a gap begins in the sequence of squarefree numbers (A005117).

Original entry on oeis.org

4, 8, 12, 16, 18, 20, 24, 27, 32, 36, 40, 44, 48, 52, 54, 56, 60, 63, 68, 72, 75, 80, 84, 88, 90, 92, 96, 98, 104, 108, 112, 116, 120, 124, 128, 132, 135, 140, 144, 147, 150, 152, 156, 160, 162, 164, 168, 171, 175, 180, 184, 188, 192, 196, 198, 200, 204, 207, 212

Offset: 1

N. J. A. Sloane, Apr 07 2000

			The first gap is at 4 and has length 1; the next starts at 8 and has length 2 (since neither 8 nor 9 are squarefree).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
M. Filaseta and O. Trifonov, On Gaps between Squarefree Numbers. In Analytic Number Theory, Vol 85, 1990, Birkhäuser, Basel, pp. 235-253.
E. Fogels, On the average values of arithmetic functions, Proc. Cambridge Philos. Soc. 1941, 37: 358-372.
L. Marmet, First occurrences of squarefree gaps...
L. Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012.
K. F. Roth, On the gaps between squarefree numbers, J. London Math. Soc. 1951 (2) 26:263-268.

Cf. A005117, A053797, A045882, A051681, A013929.

is(n)=!issquarefree(n) && issquarefree(n-1) \\ Charles R Greathouse IV, Nov 05 2017

list(lim)=my(v=List(),t); forfactored(n=4,lim\1, if(vecmax(n[2][,2])>1, if(!t, listput(v,n[1])); t=1, t=0)); Vec(v) \\ Charles R Greathouse IV, Nov 05 2017

A045882 Smallest term of first run of (at least) n consecutive integers which are not squarefree.

Original entry on oeis.org

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

Offset: 1

Erich Friedman

Solution for n=10 is same as for n=11.

This sequence is infinite and each term initiates a suitable arithmetic progression with large differences like squares of primorials or other suitable products of primes from prime factors being on power 2 in terms and in chains after. Proof includes solution of linear Diophantine equations and math. induction. See also A068781, A070258, A070284, A078144, A049535, A077640, A077647, A078143 of which first terms are recollected here. - Labos Elemer, Nov 25 2002

			a(3) = 48 as 48, 49 and 50 are divisible by squares.
n=5 -> {844=2^2*211; 845=5*13^2; 846=2*3^2*47; 847=7*11^2; 848=2^4*53}.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 242, p. 67, Ellipses, Paris 2008.

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 numbers
Eric Weisstein's World of Mathematics, Squareful

Cf. A013929, A053806, A049535, A077647, A078143. Also A069021 and A051681 are different versions.

cnt = 0; k = 0; Table[While[cnt < n, k++; If[! SquareFreeQ[k], cnt++, cnt = 0]]; k - n + 1, {n, 7}]

a(n)=my(s);for(k=1,9^99,if(issquarefree(k),s=0,if(s++==n,return(k-n+1)))) \\ Charles R Greathouse IV, May 29 2013

a(n) = 1 + A020754(n+1). - R. J. Mathar, Jun 25 2010
Correction from Jeppe Stig Nielsen, Mar 05 2022: (Start)
a(n) = 1 + A020754(n+1) for 1 <= n < 11.
a(n) = 1 + A020754(n) for 11 <= n < N where N is unknown.
Possibly a(n) = 1 + A020754(n-d) for some higher n, depending on how many repeated terms the sequence has. (End)
a(n) <= A061742(n) = A002110(n)^2 is the trivial bound obtained from the CRT. - Charles R Greathouse IV, Sep 06 2022

a(9)-a(11) from Patrick De Geest, Nov 15 1998, Jan 15 1999
a(12)-a(15) from Louis Marmet (louis(AT)marmet.org) and David Bernier (ezcos(AT)yahoo.com), Nov 15 1999
a(16) was obtained as a result of a team effort by Z. McGregor-Dorsey et al. [Louis Marmet (louis(AT)marmet.org), Jul 27 2000]
a(17) was obtained as a result of a team effort by E. Wong et al. [Louis Marmet (louis(AT)marmet.org), Jul 13 2001]
a(18) was obtained as a result of a team effort by L. Marmet et al.

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]}]

A020754 Increasing gaps between squarefree numbers (lower end).

Original entry on oeis.org

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

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

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

A077640 Smallest term of a run of at least 7 consecutive integers which are not squarefree.

Original entry on oeis.org

217070, 671346, 826824, 1092747, 1092748, 1427370, 2097048, 2779370, 3112819, 3306444, 3597723, 3994820, 4063774, 4442874, 4630544, 4842474, 5436375, 5479619, 5610644, 5634122, 6315019, 6474220, 6626319, 6677864, 7128471, 7216618, 7216619, 7295448, 7507923

Offset: 1

Labos Elemer, Nov 14 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 (terms 1..488 from Robert Israel)

Cf. A013929, A045882, A049535, A051681, A188347.

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).

N = 10^7; % to get all terms <= N-6
T = zeros(1,N);
for m = 2:floor(sqrt(N))
   T([m^2 : m^2 : N]) = 1;
end
S = T(1:N-6).*T(2:N-5).*T(3:N-4).*T(4:N-3).*T(5:N-2).*T(6:N-1).*T(7:N);
find(S)  % Robert Israel, Feb 03 2016

s7[x_] := Apply[Plus, Table[Abs[MoebiusMu[x+j]], {j, 0, 6}]]; Do[If[s7[n] == 0, Print[n]], {n, 10^7}]
Flatten[Position[Partition[SquareFreeQ/@Range[7000000],7,1],?(Union[#] == {False}&),{1},Heads->False]] (* _Harvey P. Dale, May 24 2014 *)
SequencePosition[Table[If[SquareFreeQ[n],0,1],{n,72*10^5}],{1,1,1,1,1,1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 15 2017 *)

{my(N=10^6, M=0, t, m2); for(m=2,sqrtint(N), t=1; m2=m^2; M=bitor(sum(i=1,N\m^2,t<<=m2),M)); for(i=1,6,M=bitand(M,M>>1)); for(i=0,N,M||break;print1(i+=t=valuation(M,2),",");M>>=t+1)} \\ Works but is much slower than the following (16s for 10^6 vs. 3s for 10^7). Should scale better (~sqrt(n) vs linear) but doesn't because of inefficient implementation of binary operations (copies & re-allocation of very large bitmaps): increasing N from 10^5 to 10^6 multiplies CPU time by a factor of 100!

for(n=1,10^7,forstep(k=6,0,-1,issquarefree(n+k)&&(n+=k)&&next(2));print1(n",")) \\ M. F. Hasler, Feb 03 2016

A077640 = { A078144[k] | A078144[k+2] = A078144[k]+2 } = { A070284[k] | A070284[k+3] = A070284[k]+3 } etc. Note that A049535 is defined differently. - M. F. Hasler, Feb 01 2016

a(n) = A188347(n) - 3. - Amiram Eldar, Feb 09 2021

A020755 Increasing gaps between squarefree numbers (upper end).

Original entry on oeis.org

2, 5, 10, 51, 246, 849, 22026, 217077, 1092755, 8870033, 221167433, 47255689927, 82462576233, 1043460553378, 79180770078563, 3215226335143234, 23742453640900989, 125781000834058586

Offset: 1

David W. Wilson

Up to n=10, a(n) is the upper end of the first gap of length n. However, for n=11 through n=16, a(n) is the upper end of the first gap of length n+1. See A020753. - M. F. Hasler, Dec 28 2015

			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. We are only interested in gaps that set new records.

Cf. A005117, A020754, A020753, A045882, A051681.

A020755(n)=for(k=L=1,oo,issquarefree(k)||next;k-L>=n&&return(k);L=k) \\ M. F. Hasler, Dec 28 2015

a(n) = A020754(n) + A020753(n). - M. F. Hasler, Dec 28 2015

Thanks to Christian Bower for additional comments.

More terms (computed using data from A020754) added by M. F. Hasler, Dec 28 2015

cp's OEIS Frontend

A055556 Differences of arithmetic progressions of which the terms give chains of n consecutive nonsquarefree numbers if started with terms of A045882(n) or A051681(n).

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A053806 Numbers where a gap begins in the sequence of squarefree numbers (A005117).

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A045882 Smallest term of first run of (at least) n consecutive integers which are not squarefree.

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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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A020754 Increasing gaps between squarefree numbers (lower end).

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

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

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