A188296 -id:A188296 - cp's OEIS frontend

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

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

A258332 Numbers n such that 4n + 1, 4n + 2 and 4n + 3 are not squarefree.

Original entry on oeis.org

211, 420, 722, 906, 2731, 3687, 3962, 4351, 4985, 5505, 5656, 5818, 6162, 6443, 7337, 7562, 7731, 8293, 9175, 9312, 9681, 9861, 10118, 11343, 11918, 11931, 11956, 12093, 12372, 13646, 13756, 13862, 14280, 14618, 14712, 14981, 15306, 15716, 15743, 15961, 16512, 17162, 17237

Offset: 1

Juri-Stepan Gerasimov, May 26 2015

			211 is in this sequence because 4 * 211 + 1 = 845 = 5 * 13^2, 4 * 211 + 2 = 846 = 2 * 3^2 * 47 and 4 * 211 + 3 = 847 = 7 * 11^2.

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

Cf. A188296, A258211.

[n: n in [1..20000] | not IsSquarefree(4*n+1) and  not IsSquarefree(4*n+2) and not IsSquarefree(4*n+3)];

remove(t->ormap(numtheory:-issqrfree,[4*t+1,4*t+2,4*t+3]), [$1..2*10^4]); # Robert Israel, Apr 03 2018

Select[Range[1000], Union[{MoebiusMu[4# + 1], MoebiusMu[4# + 2], MoebiusMu[4# + 3]}] == {0} &] (* Alonso del Arte, May 26 2015 *)

isok(n) = !issquarefree(4*n+1) && !issquarefree(4*n+2) && !issquarefree(4*n+3); \\ Michel Marcus, Apr 04 2018

A188347 Central element of a series of 7 successive nonsquarefree numbers.

Original entry on oeis.org

217073, 671349, 826827, 1092750, 1092751, 1427373, 2097051, 2779373, 3112822, 3306447, 3597726, 3994823, 4063777, 4442877, 4630547, 4842477, 5436378, 5479622, 5610647, 5634125, 6315022, 6474223, 6626322, 6677867, 7128474, 7216621, 7216622, 7295451, 7507926

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 29 2011

nonn

Each of a(n), a(n)-1, a(n)-2, a(n)-3, a(n)+1, a(n)+2, a(n)+3 has at least one square divisor.

			217073=17*113^2, 217073-1=2^4*13567, 217073-2=3^2*89*271, 217073-3=2*5*7^2*443, 217073+1=2*3*11^2*13*23, 217073+2=5^2*19*457, 217073+3=2^2*54269.

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

Cf. A005117, A013929, A077640, A188296.

Select[Range[200000,5000000], !SquareFreeQ[#] &&!SquareFreeQ[#-1] &&!SquareFreeQ[#+1] &&!SquareFreeQ[#-2] &&!SquareFreeQ[#+2] &&!SquareFreeQ[#-3] &&!SquareFreeQ[#+3]&]

a(n) = A077640(n) + 3. - Amiram Eldar, Feb 09 2021

More terms from Carl Najafi, Aug 22 2011

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

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A258332 Numbers n such that 4n + 1, 4n + 2 and 4n + 3 are not squarefree.

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A188347 Central element of a series of 7 successive nonsquarefree numbers.

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