A337030 -id:A337030 - cp's OEIS frontend

A378040 Union of A377783(n) = least nonsquarefree number > prime(n).

4, 8, 12, 16, 18, 20, 24, 32, 40, 44, 48, 54, 60, 63, 68, 72, 75, 80, 84, 90, 98, 104, 108, 112, 116, 128, 132, 140, 150, 152, 160, 164, 168, 175, 180, 184, 192, 196, 198, 200, 212, 224, 228, 232, 234, 240, 242, 252, 260, 264, 270, 272, 279, 284, 294, 308, 312

Offset: 1

Gus Wiseman, Nov 20 2024

nonn

Numbers k such that, if p is the greatest prime < k, all numbers from p to k (exclusive) are squarefree.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

For squarefree we have A112926 (diffs A378037), opposite A112925 (diffs A378038).
For prime-power instead of nonsquarefree we have A345531, differences A377703.
Union of A377783 (diffs A377784), restriction of A120327 (diffs A378039).
Nonsquarefree numbers not appearing are A378084, see also A378082, A378083.
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.
A070321 gives the greatest squarefree number up to n.
A071403(n) = A013928(prime(n)) counts squarefree numbers up to prime(n).
A378086(n) = A057627(prime(n)) counts nonsquarefree numbers up to prime(n).
Cf. A378034 (differences of A378032), restriction of A378036 (differences A378033).
Cf. A000015, A053797, A053806, A072284, A224363, A337030, A377047, A377049, A377430, A377431.

Union[Table[NestWhile[#+1&,Prime[n],SquareFreeQ],{n,100}]]
lns[p_]:=Module[{k=p+1},While[SquareFreeQ[k],k++];k]; Table[lns[p],{p,Prime[Range[70]]}]//Union (* Harvey P. Dale, Jun 12 2025 *)

A378084 Nonsquarefree numbers not appearing in A377783 (least nonsquarefree number > prime(n)).

Original entry on oeis.org

9, 25, 27, 28, 36, 45, 49, 50, 52, 56, 64, 76, 81, 88, 92, 96, 99, 100, 117, 120, 121, 124, 125, 126, 135, 136, 144, 147, 148, 153, 156, 162, 169, 171, 172, 176, 188, 189, 204, 207, 208, 216, 220, 225, 236, 243, 244, 245, 248, 250, 256, 261, 268, 275, 276, 280

Offset: 1

Gus Wiseman, Nov 23 2024

nonn

Warning: do not confuse with A377784.

			The terms together with their prime indices begin:
    9: {2,2}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   36: {1,1,2,2}
   45: {2,2,3}
   49: {4,4}
   50: {1,3,3}
   52: {1,1,6}
   56: {1,1,1,4}
   64: {1,1,1,1,1,1}
   76: {1,1,8}
   81: {2,2,2,2}
   88: {1,1,1,5}
   92: {1,1,9}
   96: {1,1,1,1,1,2}

Disjoint from A377783 (union A378040), first-differences A377784.
Appearing once: A378082.
Appearing twice: A378083.
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 (sums A337030), zeros A068360.
A061399 counts nonsquarefree numbers between primes (sums A378086), zeros A068361.
A070321 gives the greatest squarefree number up to n.
A112925 gives least squarefree number > prime(n), differences A378038.
A112926 gives greatest squarefree number < prime(n), differences A378037.
A120327 (union A162966) gives least nonsquarefree number >= n, differences A378039.
A377046 encodes k-differences of nonsquarefree numbers, zeros A377050.
Cf. A053797, A053806, A072284, A224363, A377430, A377431, A378032, A378033.

nn=100;
y=Table[NestWhile[#+1&,Prime[n],SquareFreeQ[#]&],{n,nn}];
Complement[Select[Range[Prime[nn]],!SquareFreeQ[#]&],y]

Complement of A378040 in A013929.

A378082 Terms appearing only once in A377783 = least nonsquarefree number > prime(n).

Original entry on oeis.org

12, 16, 18, 20, 24, 40, 48, 54, 60, 63, 68, 72, 75, 80, 84, 90, 98, 108, 112, 116, 128, 132, 150, 152, 160, 164, 168, 175, 180, 184, 192, 196, 198, 200, 212, 224, 228, 232, 234, 240, 242, 252, 260, 264, 270, 272, 279, 294, 308, 312, 315, 320, 332, 338, 348

Offset: 1

Gus Wiseman, Nov 20 2024

nonn

Nonsquarefree numbers k such that if p < q are the two greatest primes < k, there is at least one nonsquarefree number between p and q but all numbers between q and k are squarefree. - Robert Israel, Nov 20 2024

			The terms together with their prime indices begin:
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   60: {1,1,2,3}
   63: {2,2,4}
   68: {1,1,7}
   72: {1,1,1,2,2}
   75: {2,3,3}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   90: {1,2,2,3}
   98: {1,4,4}
  108: {1,1,2,2,2}
  112: {1,1,1,1,4}
  116: {1,1,10}
  128: {1,1,1,1,1,1,1}
  132: {1,1,2,5}

This is a transformation of A377783 (union A378040, differences A377784).
Note also A377783 restricts A120327 (differences A378039) to the primes.
Terms appearing twice are A378083.
Terms not appearing at all are A378084.
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.
A070321 gives the greatest squarefree number up to n.
A071403(n) = A013928(prime(n)) counts squarefree numbers < prime(n).
A378086(n) = A057627(prime(n)) counts nonsquarefree numbers < prime(n).
Cf. A112926 (diffs A378037), opposite A112925 (diffs A378038).
Cf. A378032 (diffs A378034), restriction of A378033 (diffs A378036).
Cf. A053797, A053806, A072284, A112929, A120992, A224363, A337030, A377430, A377431, A377703.

q:= 3: R:= NULL: flag:= false: count:= 0:
while count < 100 do
  p:= q; q:= nextprime(q);
  for k from p+1 to q-1 do
    found:= false;
    if not numtheory:-issqrfree(k) then
      if flag then
          count:= count+1; R:= R,k
      fi;
      found:= true; break
    fi;
   od;
   flag:= found;
od:
R; # Robert Israel, Nov 20 2024

y=Table[NestWhile[#+1&,Prime[n],SquareFreeQ],{n,100}];
Select[Most[Union[y]],Count[y,#]==1&]

A378083 Nonsquarefree numbers appearing exactly twice in A377783 (least nonsquarefree number > prime(n)).

Original entry on oeis.org

4, 8, 32, 44, 104, 140, 284, 464, 572, 620, 644, 824, 860, 1232, 1292, 1304, 1484, 1700, 1724, 1880, 2084, 2132, 2240, 2312, 2384, 2660, 2732, 2804, 3392, 3464, 3560, 3920, 3932, 4004, 4220, 4244, 4424, 4640, 4724, 5012, 5444, 5480, 5504, 5660, 6092, 6200

Offset: 1

Gus Wiseman, Nov 23 2024

nonn

Warning: do not confuse with A377783.

			The terms together with their prime indices begin:
     4: {1,1}
     8: {1,1,1}
    32: {1,1,1,1,1}
    44: {1,1,5}
   104: {1,1,1,6}
   140: {1,1,3,4}
   284: {1,1,20}
   464: {1,1,1,1,10}
   572: {1,1,5,6}
   620: {1,1,3,11}
   644: {1,1,4,9}
   824: {1,1,1,27}
   860: {1,1,3,14}
  1232: {1,1,1,1,4,5}

Subset of A377783 (union A378040, diffs A377784), restriction of A120327 (diffs A378039).
Terms appearing once are A378082.
Terms not appearing at all are A378084.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
A071403(n) = A013928(prime(n)) counts squarefree numbers < prime(n).
A378086(n) = A057627(prime(n)) counts nonsquarefree numbers < prime(n).
Cf. A112926 (diffs A378037), opposite A112925 (diffs A378038).
Cf. A378032 (diffs A378034), restriction of A378033 (diffs A378036).
Cf. A053797, A053806, A070321, A072284, A112929, A120992, A224363, A337030, A377430, A377431, A377703.

y=Table[NestWhile[#+1&,Prime[n],SquareFreeQ[#]&],{n,1000}];
Select[Union[y],Count[y,#]==2&]

A380413 Terms appearing twice in A378086 (number of nonsquarefree numbers < prime(n)).

Original entry on oeis.org

0, 1, 11, 14, 39, 53, 109, 179, 222, 240, 251, 319, 337, 481, 505, 508, 578, 664, 674, 738, 818, 835, 877, 905, 933, 1041, 1069, 1098, 1325, 1352, 1392, 1535, 1539, 1567, 1652, 1663, 1732, 1817, 1849, 1960, 2134, 2148, 2158, 2220, 2387, 2428, 2457, 2622, 2625

Offset: 1

Gus Wiseman, Feb 06 2025

nonn

A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A061399 counts nonsquarefree integers between primes, see A068361, A061398, A068360, A377783, A378086.
A070321 gives the greatest squarefree number up to n.
A071403 counts squarefree numbers < prime(n), see A373198, A337030.
A112925 gives the greatest squarefree number between primes, least A112926.
Cf. A057627, A065890, A378032 (differences A378034), A378033 (differences A378036).
Cf. A072284, A120327, A224363, A378040, A378082, A378084.

y=Table[Length[Select[Range[Prime[n]],!SquareFreeQ[#]&]],{n,100}];
Select[Most[Union[y]],Count[y,#]==2&]

a(n) = A378086(A068361(n)) = A378086(A068361(n)+1).

cp's OEIS Frontend

A378040 Union of A377783(n) = least nonsquarefree number > prime(n).

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A378084 Nonsquarefree numbers not appearing in A377783 (least nonsquarefree number > prime(n)).

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A378082 Terms appearing only once in A377783 = least nonsquarefree number > prime(n).

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A378083 Nonsquarefree numbers appearing exactly twice in A377783 (least nonsquarefree number > prime(n)).

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A380413 Terms appearing twice in A378086 (number of nonsquarefree numbers < prime(n)).

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