A070284 -id:A070284 - cp's OEIS frontend

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

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.

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

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

A175590 Numbers k with prime signature(k) = prime signature(k+1) = prime signature(k+2) = prime signature(k+3).

Original entry on oeis.org

19940, 49147, 54585, 118923, 136825, 183554, 204323, 204324, 262932, 304675, 361275, 361322, 476377, 486962, 506905, 619722, 668211, 734948, 854018, 937025, 938203, 999649, 1062025, 1118275, 1335572, 1336075, 1356324, 1466225, 1541491

Offset: 1

Charles R Greathouse IV, Jul 19 2010

nonn

			a(1) = 2^2 * 5 * 997; a(1)+1 = 3 * 17^2 * 23; a(1)+2 = 2 * 13^2 * 59; a(1)+3 = 7^2 * 11 * 37. All have prime signature {2, 1, 1}.

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

Cf. A052213, A052214, A218448. Subsequence of A070284.

SequencePosition[Table[Sort[FactorInteger[n][[All,2]]],{n,1542000}],{x_,x_,x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* The program will take a long time to run. *) (* Harvey P. Dale, Jun 09 2021 *)

sig(n)={vecsort(factor(n)[,2])}; s=sig(1);for(n=1,1e6,t=sig(n+1);if(s==t&t==sig(n+2)&t==sig(n+3),print1(n-1,","));s=t)

is_A175590(n)={my(f(n)=vecsort(factor(n)[,2]),t=f(n));!for(i=1,3,f(n+i)!=t & return)}  \\ M. F. Hasler, Nov 01 2012

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

A114180 Numbers n with mu(n) = mu(n+1) = mu(n+2).

Original entry on oeis.org

29, 33, 41, 48, 85, 93, 98, 101, 124, 137, 141, 201, 213, 217, 229, 242, 243, 281, 301, 342, 350, 393, 423, 429, 433, 445, 475, 548, 603, 617, 633, 641, 645, 697, 724, 741, 774, 821, 844, 845, 846, 869, 921, 969, 1021, 1024, 1041, 1085, 1129, 1137, 1189

Offset: 1

Franklin T. Adams-Watters, Feb 03 2006

nonn

Any sequence of 4 or more consecutive numbers with the same value for mu must all have mu(n)=0 (n divisible by a proper square) since at least one of every 4 consecutive numbers is divisible by 4.

A261890(a(n)) = 0. - Reinhard Zumkeller, Sep 05 2015

			mu(n)=1 for 33,34,35; 85,86,87; 93,94,95; ...
mu(n)=-1 for 29,30,31; 41,42,43; 101,102,103; ...
mu(n)=0 for 48,49,50; 98,99,100; 124,125,126; ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Union of A070258, A063838 and A070268. Cf. A008683, A070284.

Cf. A261890.

a114180 n = a114180_list !! (n-1)
a114180_list = filter ((== 0) . a261890) [1..]
-- Reinhard Zumkeller, Sep 05 2015

SequencePosition[MoebiusMu[Range[1200]],{x_,x_,x_}][[;;,1]] (* Harvey P. Dale, Jul 23 2023 *)

cp's OEIS Frontend

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

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A049535 Starts of runs of exactly 6 consecutive nonsquarefree numbers.

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

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

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

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

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A175590 Numbers k with prime signature(k) = prime signature(k+1) = prime signature(k+2) = prime signature(k+3).

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