A073247 -id:A073247 - cp's OEIS frontend

A379309 Number of strict integer partitions of n with a unique squarefree part.

0, 1, 1, 1, 0, 2, 2, 2, 0, 2, 4, 4, 1, 4, 7, 7, 2, 6, 8, 11, 4, 9, 13, 17, 7, 13, 20, 22, 13, 20, 29, 33, 21, 29, 40, 47, 27, 41, 56, 64, 42, 59, 77, 85, 60, 74, 104, 115, 83, 101, 141, 155, 113, 138, 179, 206, 156, 183, 236, 272, 212, 239, 309, 343, 282, 315

Offset: 0

Gus Wiseman, Dec 27 2024

nonn

			The a(9) = 2 through a(15) = 7 partitions:
  (5,4)  (10)   (11)   (9,3)  (13)     (14)     (15)
  (8,1)  (6,4)  (7,4)         (8,5)    (8,6)    (8,7)
         (8,2)  (8,3)         (12,1)   (9,5)    (9,6)
         (9,1)  (9,2)         (8,4,1)  (10,4)   (11,4)
                                       (12,2)   (12,3)
                                       (8,4,2)  (8,4,3)
                                       (9,4,1)  (9,4,2)

If all parts are squarefree we have A087188, non-strict A073576 (ranks A302478).
If no parts are squarefree we have A256012, non-strict A114374 (ranks A379307).
For composite instead of squarefree we have A379303, non-strict A379302 (ranks A379301).
For prime instead of squarefree we have A379305, non-strict A379304 (ranks A331915).
The non-strict version is A379308, ranks A379316.
For old prime instead of squarefree we have A379315, non-strict A379314 (ranks A379312).
Ranked by A379316 /\ A005117 = squarefree positions of 1 in A379306.
A000041 counts integer partitions, strict A000009.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A377038 gives k-th differences of squarefree numbers.
A379310 counts nonsquarefree prime indices.
Cf. A000586, A000607, A002095, A023895, A034891, A036497, A072284, A073247, A096258, A204389, A377430.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?SquareFreeQ]==1&]],{n,0,30}]

lista(nn) = my(r=1, s=0); for(k=1, nn, if(issquarefree(k), s+=x^k, r*=1+x^k)); concat(0, Vec(r*s+O(x^(1+nn)))); \\ Jinyuan Wang, Feb 21 2025

More terms from Jinyuan Wang, Feb 21 2025

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

Original entry on oeis.org

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}]]

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}]]

A073248 Squarefree numbers k such that k+1 is also squarefree, but k-1 and k+2 are not.

Original entry on oeis.org

10, 46, 61, 73, 82, 118, 122, 133, 145, 154, 173, 190, 205, 226, 246, 262, 273, 277, 290, 298, 313, 326, 334, 370, 373, 385, 406, 421, 426, 442, 457, 473, 478, 493, 505, 514, 526, 537, 565, 573, 586, 601, 606, 622, 626, 658, 673, 694, 709, 730, 733, 745

Offset: 1

Reinhard Zumkeller, Jul 22 2002

nonn

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

Cf. A005117, A073250, A073247, A007675.

state:= [true,true,true,true]:
R:= NULL: count:= 0:
for n from 1 while count < 100 do
  state:= [state[2],state[3],state[4],numtheory:-issqrfree(n)];
  if state = [false,true,true,false] then
     R:= R, n-2; count:= count+1
  fi
od:
R; # Robert Israel, Mar 02 2022

Transpose[SequencePosition[Table[If[SquareFreeQ[n],1,0],{n,800}],{0,1,1,0}]][[1]]+1 (* The program uses the SequencePosition function from Mathematica version 10 *) (* Harvey P. Dale, Mar 09 2016 *)

A268332 Squarefree numbers differing by more than 3 from any other squarefree number.

Original entry on oeis.org

2526, 44405, 47527, 47973, 55779, 72474, 101037, 106327, 106674, 109023, 110107, 133577, 153173, 165574, 183553, 186247, 193026, 196747, 208847, 209674, 212127, 218527, 220209, 234622, 237522, 245149, 261478, 266853, 269953, 308649, 328877, 334522, 342066, 364151, 370785, 375823

Offset: 1

Christopher E. Thompson, Feb 01 2016

nonn

Christopher E. Thompson, Table of n, a(n) for n = 1..1000

Cf. A005117, A073247, A268331, A268333, A268334.

SF:= select(numtheory:-issqrfree, [$1..10^6]):
SF[select(i -> SF[i]-SF[i-1]>=4  and SF[i+1]-SF[i]>=4, [$2..nops(SF)-1])]; # Robert Israel, Feb 02 2016

A268334 Squarefree numbers differing by more than 5 from any other squarefree number.

Original entry on oeis.org

8061827, 60529549, 82490423, 213819827, 245990821, 360350923, 364661627, 465966527, 494501773, 531794155, 651012673, 661327077, 665519027, 693770521, 765921647, 800254429, 826112857, 836818573, 885970253, 914588627, 930996623, 936321349, 958710973, 992890427, 1069677149

Offset: 1

Christopher E. Thompson, Feb 01 2016

nonn

Christopher E. Thompson, Table of n, a(n) for n = 1..100

Cf. A073247, A268331, A268332, A268333.

A268330 Least squarefree number differing by more than n from any other squarefree number.

Original entry on oeis.org

1, 17, 26, 2526, 5876126, 8061827, 8996188226, 2074150570370

Offset: 0

Christopher E. Thompson, Feb 01 2016

1.8*10^12 < a(7) <= 10735237201449 - Robert Israel, Mar 18 2016

a(8) > 5*10^12. - Giovanni Resta, Apr 11 2016

			a(2) = 26 because 26 is squarefree but 24,25,27,28 are not.

Cf. A073247, A268331, A268332, A268333, A268334.

B = 10^8; % blocks of size B
nB = 1000; % nB blocks
A = [1];
P = primes(floor(sqrt(nB*B)));
mmax = 1;
i0 = 1;
for k = 0:nB-1  % search squarefrees from i0+1 to i0 + B
  V = true(1, B);
  for i = 1:numel(P)
    p = P(i);
    V([(p^2 - mod(i0,p^2)):p^2:B]) = false;
  end
  SF = find(V) + i0;
  DSF = SF(2:end) - SF(1:end-1);
  i0 = SF(end-2);
  M = min(DSF(1:end-1), DSF(2:end));
  newmax = max(mmax,max(M));
  for i = mmax+1:newmax
    A(i) = SF(1 + find(M>=i, 1, 'first'));
  end
  mmax = newmax;
end
for i=1:mmax
  fprintf('%d ',A(i));
end
fprintf('\n');  % Robert Israel, Mar 16 2016

(* implementation assumes a(n) is increasing *)
nsfRun[n_]:=Module[{i=n}, While[!SquareFreeQ[i], i++]; i-n]
a268330[{low_, high_}, width_]:=Module[{k=width, i, next, r, s, list={}}, For[i=low, i<=high, i+=next, r=nsfRun[i]; If[r0 (* Hartmut F. W. Hoft, Mar 15 2016 *)
a268330[{0,10000000},1] (* computes a(1)...a(5) *)

a(6) from Hartmut F. W. Hoft, Mar 15 2016

a(7) from Giovanni Resta, Apr 11 2016

A268331 Squarefree numbers differing by more than 2 from any other squarefree number.

Original entry on oeis.org

26, 170, 362, 530, 638, 727, 874, 926, 962, 1027, 1126, 1423, 1574, 1774, 1814, 1826, 1861, 2059, 2402, 2526, 2674, 2726, 2782, 2874, 3178, 3482, 3574, 3719, 3774, 3970, 4166, 4474, 4490, 4526, 4654, 5042, 5045, 5374, 5426, 5914, 5930, 6026, 6173, 6254, 6274, 6326, 6418, 6626, 6649, 6726, 7138, 7174

Offset: 1

Christopher E. Thompson, Feb 01 2016

nonn

			26 is a term because 26 is squarefree but 24,25,27,28 are not.

Christopher E. Thompson, Table of n, a(n) for n = 1..1000

Cf. A073247, A268332, A268333, A268334.

SequencePosition[Table[If[SquareFreeQ[n],1,0],{n,7200}],{0,0,1,0,0}][[All,1]]+2 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 19 2018 *)

A268333 Squarefree numbers differing by more than 4 from any other squarefree number.

Original entry on oeis.org

5876126, 8061827, 19375679, 27926071, 29002021, 29850943, 39224453, 39728861, 54427974, 56389147, 60529549, 63520174, 67806346, 71987374, 75239979, 82490423, 87011827, 91332377, 97447622, 99368949, 100842249, 107447838, 110196277, 118389143, 134817726, 149840974, 159140777, 161651279

Offset: 1

Christopher E. Thompson, Feb 01 2016

nonn

Christopher E. Thompson, Table of n, a(n) for n = 1..200

Cf. A073247, A268331, A268332, A268334.

A378373 Number of composite numbers (A002808) between consecutive nonsquarefree numbers (A013929), exclusive.

Original entry on oeis.org

1, 0, 1, 2, 0, 0, 2, 0, 1, 0, 1, 3, 2, 1, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 2, 2, 1, 0, 2, 0, 1, 3, 0, 1, 3, 0, 0, 0, 1, 2, 2, 2, 0, 2, 0, 2, 0, 0, 0, 2, 2, 0, 1, 3, 2, 0, 0, 0, 0, 2, 2, 1, 0, 2, 0, 1, 0, 1, 0, 2, 2, 3, 0, 1, 2, 0, 0, 3, 2, 0, 2, 3, 3, 2, 0, 1, 2

Offset: 1

Gus Wiseman, Dec 02 2024

nonn

All terms are 0, 1, 2, or 3 (cf. A078147).
The inclusive version is a(n) + 2.
The nonsquarefree numbers begin: 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, ...

			The composite numbers counted by a(n) form the following set partition of A120944:
{6}, {}, {10}, {14,15}, {}, {}, {21,22}, {}, {26}, {}, {30}, {33,34,35}, {38,39}, ...

For prime (instead of nonsquarefree) we have A046933.
For squarefree (instead of nonsquarefree) we have A076259(n)-1.
For prime power (instead of nonsquarefree) we have A093555.
For prime instead of composite we have A236575.
For nonprime prime power (instead of nonsquarefree) we have A378456.
For perfect power (instead of nonsquarefree) we have A378614, primes A080769.
A002808 lists the composite numbers.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A073247 lists squarefree numbers with nonsquarefree neighbors.
A120944 lists squarefree composite numbers.
A377432 counts perfect-powers between primes, zeros A377436.
A378369 gives distance to the next nonsquarefree number (A120327).
Cf. A065890, A067535, A067871, A080101, A081221, A151800, A243348, A366833, A377046, A378033, A378039, A378086.

v=Select[Range[100],!SquareFreeQ[#]&];
Table[Length[Select[Range[v[[i]]+1,v[[i+1]]-1],CompositeQ]],{i,Length[v]-1}]

cp's OEIS Frontend

A379309 Number of strict integer partitions of n with a unique squarefree part.

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

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Mathematica

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

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Mathematica

A073248 Squarefree numbers k such that k+1 is also squarefree, but k-1 and k+2 are not.

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Maple

Mathematica

A268332 Squarefree numbers differing by more than 3 from any other squarefree number.

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Maple

A268334 Squarefree numbers differing by more than 5 from any other squarefree number.

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A268330 Least squarefree number differing by more than n from any other squarefree number.

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MATLAB

Mathematica

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A268331 Squarefree numbers differing by more than 2 from any other squarefree number.

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Mathematica

A268333 Squarefree numbers differing by more than 4 from any other squarefree number.

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A378373 Number of composite numbers (A002808) between consecutive nonsquarefree numbers (A013929), exclusive.

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