A045882 -id:A045882 - cp's OEIS frontend

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

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

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}]]

A330486 Earliest start of a run of n numbers divisible by a seventh power larger than one.

Original entry on oeis.org

128, 76544, 2372890624, 390491792890623, 2083234733888734218749, 18962123650219836035505781245

Offset: 1

Jud McCranie, Dec 15 2019

In De Koninck's book, a(4) was probable but not certain.  It also gives probable terms for a(5) and a(6).
From David A. Corneth, Dec 18 2019: (Start)
a(6) <= 18962123650219836035505781245
a(7) <= 105574810222799317012520534891328125
a(8) <= 352061173826332779662070232460962002804453118
a(9) <= 74322635218313783849790472504951802188239215127109371
If any a(k) of these values are less than presented values, one of a(k) + m is divisible only by at least one p^7 for some p > 31, 0 <= m < k.
For a(6) this minimum prime is some prime p > 59. (End)

			2372890624 is divisible by 2^7, 2372890625 is divisible by 5^7, 2372890626 is divisible by 3^7.  This is the smallest number with this property, so a(3)=2372890624.

J.-M. De Koninck, Those Fascinating Numbers, Entry 242, p. 63, Amer. Math. Soc., 2009.

David A. Corneth, PARI program for finding candidates
Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., (2009), page 63.

Cf. A045882, A271443, A001015.

a(5) from Giovanni Resta, Dec 17 2019

a(6) from Giovanni Resta, Dec 19 2019

A330483 Earliest start of a run of n numbers divisible by an eighth power larger than one.

Original entry on oeis.org

256, 636416, 70925781248, 36430999887109373, 5031679407516945387109374

Offset: 1

Jud McCranie, Dec 16 2019

Similar to sequences in the crossreferences.

a(6) <= 330987813033061096652260864453120. - Robert Israel, Jun 02 2020

			70925781248 is divisible by 2^8, 70925781249 is divisible by 3^8, 70925781250 is divisible by 5^8. This is the smallest number with this property, so a(3) = 70925781248.

Cf. A045882, A271443.

a(4) from Giovanni Resta, Dec 16 2019

a(5) from Giovanni Resta, Dec 17 2019

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

A268313 First term of a run of at least 10 consecutive integers which are not squarefree.

Original entry on oeis.org

221167422, 221167423, 262315467, 363504972, 463239475, 1202800371, 1407472722, 1407472723, 1557947844, 1609077723, 1911823144, 2217728772, 2695179044, 2737800168, 2847305571, 3639720042, 3639720043, 3672883247, 3865964268, 3865964269, 3982659575, 4246929267, 4818537743, 4982931368

Offset: 1

M. F. Hasler, Feb 01 2016

nonn

M. F. Hasler, Table of n, a(n) for n = 1..61

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), A268314 (11-chains).

s10[x_] := Apply[Plus, Table[Abs[MoebiusMu[x+j]], {j, 0, 9}]]; Do[If[Equal[s10[n], 0], Print[n]], {n, 10^8, 10^10}]

A268313 = { A078143[k] | A078143[k+1] = A078143[k]+1 } = { A077647[k] | A077647[k+2] = A077647[k]+2 } = { A077640[k] | A077640[k+3] = A077640[k]+3 }.

A268314 First term of a run of at least 11 consecutive integers which are not squarefree.

Original entry on oeis.org

221167422, 1407472722, 3639720042, 3865964268, 4982931368, 5005996146, 7108776620, 8044261244, 10249558974, 12766690268, 13585489166, 19792784322, 26995377572, 30410811296, 30477326444, 32070270968, 34317891368, 39956560824, 40841363528, 42216508746, 43133805944, 46295514872, 47255689915

Offset: 1

M. F. Hasler, Feb 01 2016

nonn

a(23) is the first term beginning a 12-chain. - Bill Hannaford, Oct 06 2016

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).

s11[x_] := Apply[Plus, Table[Abs[MoebiusMu[x+j]], {j, 0, 10}]]; Do[If[Equal[s11[n], 0], Print[n]], {n, 10^8, 10^13}]

A268314 = { A268313[k] | A268313[k+1] = A268313[k]+1 } = { A078143[k] | A078143[k+2] = A078143[k]+2 } = { A077647[k] | A077647[k+3] = A077647[k]+3 } = { A077640[k] | A077640[k+4] = A077640[k]+4 }.

a(12)-a(23) from Bill Hannaford, Oct 06 2016

A330480 Earliest start of a run of n numbers divisible by a fourth power larger than one.

Original entry on oeis.org

16, 80, 33614, 202099373, 40280549372, 430995495889374, 77405340617896874

Offset: 1

Jud McCranie, Dec 16 2019

De Koninck's book says that "most likely" a(7)=77405340617896874.

			33614 is divisible by 7^4, 33615 is divisible by 3^4, and 33616 is divisible by 2^4.  This is the smallest number with this property, so a(3)=33614.

J.-M. De Koninck, Those Fascinating Numbers, Entry 242, p. 63, Amer. Math. Soc., 2009.

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

Cf. A088080, A174113.

Cf. A045882, A271443, A330481, A330482, A330486, A330483, A330484, A330485.

a(7) from Giovanni Resta, Dec 18 2019

A330481 Earliest start of a run of n numbers divisible by a fifth power larger than one.

Original entry on oeis.org

32, 1215, 2590623, 2146909373, 105636978090621, 3269698976575137500

Offset: 1

Jud McCranie, Dec 15 2019

De Koninck's book gives probable a(6) = 3269698976575137500.

a(7) <= 745436945776920355709374. - Giovanni Resta, Dec 19 2019

			1215 is divisible by 3^5 and 1216 is divisible by 2^5. This is the smallest number with this property, so a(2)=1215.

J.-M. De Koninck, Those Fascinating Numbers, Entry 242, p. 63, Amer. Math. Soc., 2009.

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

Cf. A000584, A045882, A271443, A330480, A330482, A330486, A330483, A330484, A330485, A088080.

a(6) from Giovanni Resta, Dec 19 2019

A330482 Earliest start of a run of n numbers divisible by a sixth power larger than one.

Original entry on oeis.org

64, 16767, 26890623, 1507545109375, 777562026420218750, 283435321166212288109372

Offset: 1

Jud McCranie, Dec 16 2019

De Konnick's book gives probable terms a(5)=777562026420218750 and a(6)=283435321166212288109372.

			26890623 is divisible by 3^6, 26890624 is divisible by 2^6, and 26890625 is divisible by 5^6.  This is the smallest number with this property, so a(3) = 26890623.

J.-M. De Koninck, Those Fascinating Numbers, Entry 242, p. 63, Amer. Math. Soc., 2009.

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

Cf. A001014, A088080, A174113.

Cf. A045882, A271443, A330480, A330481, A330486, A330483, A330484, A330485.

a(5) from Giovanni Resta, Dec 17 2019

a(6) from Giovanni Resta, Dec 19 2019

cp's OEIS Frontend

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

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

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A330486 Earliest start of a run of n numbers divisible by a seventh power larger than one.

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A330483 Earliest start of a run of n numbers divisible by an eighth power larger than one.

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A020755 Increasing gaps between squarefree numbers (upper end).

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A268313 First term of a run of at least 10 consecutive integers which are not squarefree.

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A268314 First term of a run of at least 11 consecutive integers which are not squarefree.

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A330480 Earliest start of a run of n numbers divisible by a fourth power larger than one.

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