A068781 -id:A068781 - cp's OEIS frontend

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

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

A375709 Numbers k such that A013929(k+1) = A013929(k) + 1. In other words, the k-th nonsquarefree number is 1 less than the next nonsquarefree number.

Original entry on oeis.org

2, 8, 10, 15, 17, 18, 24, 28, 30, 37, 38, 43, 45, 47, 48, 52, 56, 59, 65, 67, 69, 73, 80, 85, 92, 93, 94, 100, 106, 108, 111, 115, 122, 125, 128, 133, 134, 137, 138, 141, 143, 145, 148, 153, 158, 165, 166, 171, 178, 183, 184, 192, 196, 198, 203, 205, 207, 210

Offset: 1

Gus Wiseman, Sep 01 2024

nonn

The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
- A375709 (difference 1) (this)
- A375710 (difference 2)
- A375711 (difference 3)
- A375712 (difference 4)

			The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by one after the 2nd and 8th terms.

Positions of 1's in A078147.
For prime-powers (A246655) we have A375734.
First differences are A373409.
For prime numbers we have A375926.
For squarefree instead of nonsquarefree we have A375927.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A373410, A373573.

Join@@Position[Differences[Select[Range[100],!SquareFreeQ[#]&]],1]

Complement of A375710 U A375711 U A375712.

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

A373402 Numbers k such that the k-th maximal antirun of prime numbers > 3 has length different from all prior maximal antiruns. Sorted list of positions of first appearances in A027833.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 21, 24, 30, 35, 40, 41, 46, 50, 69, 82, 131, 140, 185, 192, 199, 210, 248, 251, 271, 277, 325, 406, 423, 458, 645, 748, 811, 815, 826, 831, 987, 1053, 1109, 1426, 1456, 1590, 1629, 1870, 1967, 2060, 2371, 2607, 2920, 2946, 3564, 3681, 4119

Offset: 1

Gus Wiseman, Jun 10 2024

nonn

The unsorted version is A373401.

For this sequence, we define an antirun to be an interval of positions at which consecutive primes differ by at least 3.

			The maximal antiruns of prime numbers > 3 begin:
    5
    7  11
   13  17
   19  23  29
   31  37  41
   43  47  53  59
   61  67  71
   73  79  83  89  97 101
  103 107
  109 113 127 131 137
  139 149
  151 157 163 167 173 179
The a(n)-th rows begin:
    5
    7  11
   19  23  29
   43  47  53  59
   73  79  83  89  97 101
  109 113 127 131 137

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

For squarefree runs we have the triple (1,3,5), firsts of A120992.
For prime runs we have the triple (1,2,3), firsts of A175632.
For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
For squarefree antiruns: A373200, unsorted A373128, firsts of A373127.
For composite runs we have A373400, unsorted A073051.
The unsorted version is A373401, firsts of A027833.
For composite antiruns we have the triple (1,2,7), firsts of A373403.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A006512, A007674, A049093, A068781, A072284, A077641, A174965, A251092, A373198, A373408, A373411.

t=Length/@Split[Select[Range[4,10000],PrimeQ],#1+2!=#2&]//Most;
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

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

A373573 Least k such that the k-th maximal antirun of nonsquarefree numbers has length n. Position of first appearance of n in A373409.

Original entry on oeis.org

6, 1, 18, 8, 4, 2, 10, 52, 678

Offset: 1

Gus Wiseman, Jun 10 2024

nonn

The sorted version is A373574.
An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.
Is this sequence finite? Are there only 9 terms?

			The maximal antiruns 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  68  72  75
  76  80
  81  84  88  90  92  96  98
  99
The a(n)-th rows are:
    49
     4    8
   148  150  152
    64   68   72   75
    28   32   36   40   44
     9   12   16   18   20   24
    81   84   88   90   92   96   98
   477  480  484  486  488  490  492  495
  6345 6348 6350 6352 6354 6356 6358 6360 6363

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

For composite runs we have A073051, firsts of A176246, sorted A373400.
For squarefree runs we have the triple (5,3,1), firsts of A120992.
For prime runs we have the triple (1,3,2), firsts of A175632.
For squarefree antiruns we have A373128, firsts of A373127, sorted A373200.
For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
For prime antiruns we have A373401, firsts of A027833, sorted A373402.
For composite antiruns we have the triple (2,7,1), firsts of A373403.
Positions of first appearances in A373409.
The sorted version is A373574.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A007674, A025157, A049094, A061399, A068781, A072284, A110969, A251092, A294242, A373410, A373412.

t=Length/@Split[Select[Range[10000],!SquareFreeQ[#]&],#1+1!=#2&]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[t,Range[#1]]&];
Table[Position[t,k][[1,1]],{k,spna[t]}]

cp's OEIS Frontend

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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A375709 Numbers k such that A013929(k+1) = A013929(k) + 1. In other words, the k-th nonsquarefree number is 1 less than the next nonsquarefree number.

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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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A373402 Numbers k such that the k-th maximal antirun of prime numbers > 3 has length different from all prior maximal antiruns. Sorted list of positions of first appearances in A027833.

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