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

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

1,1

Comments

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

Links

Table of n, a(n) for n=1..55.

Example

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}

Maple

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

Mathematica

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

Crossrefs

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.

Sequence in context: A119911 A272861 A192690 * A064499 A080258 A054290

Adjacent sequences: A378079 A378080 A378081 * A378083 A378084 A378085

Keyword

nonn

Author

Gus Wiseman, Nov 20 2024

Status

approved