A378037 -id:A378037 - cp's OEIS frontend

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

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}]]

A378618 Sum of nonsquarefree numbers between prime(n) and prime(n+1).

Original entry on oeis.org

0, 4, 0, 17, 12, 16, 18, 20, 104, 0, 68, 40, 0, 89, 199, 110, 60, 127, 68, 72, 151, 161, 172, 278, 297, 0, 104, 108, 112, 849, 128, 403, 0, 579, 150, 461, 322, 164, 680, 351, 180, 561, 192, 196, 198, 819, 648, 449, 228, 232, 470, 240, 1472, 508, 521, 532, 270

Offset: 1

Gus Wiseman, Dec 09 2024

nonn

			The nonsquarefree numbers between prime(24) = 89 and prime(25) = 97 are {90, 92, 96}, so a(24) = 278.

For prime instead of nonsquarefree we have A001043.
For composite instead of nonsquarefree we have A054265.
Zeros are A068361.
A000040 lists the primes, differences A001223, seconds A036263.
A070321 gives the greatest squarefree number up to n.
A071403 counts squarefree numbers up to prime(n), restriction of A013928.
A120327 gives the least nonsquarefree number >= n.
A378086 counts nonsquarefree numbers up to prime(n), restriction of A057627.
For squarefree numbers (A005117, differences A076259) between primes:
- length is A061398, zeros A068360
- min is A112926, differences A378037
- max is A112925, differences A378038
- sum is A373197
For nonsquarefree numbers (A013929, differences A078147) between primes:
- length is A061399
- min is A377783 (differences A377784), union A378040
- max is A378032 (differences A378034), restriction of A378033 (differences A378036)
- sum is A378618 (this)
Cf. A007674, A045882, A053806, A067535, A072284, A073247, A179211, A224363, A373198, A377049, A377431.

Table[Total[Select[Range[Prime[n],Prime[n+1]],!SquareFreeQ[#]&]],{n,100}]

cp's OEIS Frontend

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

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