A081221 -id:A081221 - cp's OEIS frontend

A378369 Distance between n and the least nonsquarefree number >= n.

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

Offset: 1

Gus Wiseman, Dec 01 2024

nonn

All terms are 0, 1, 2, or 3 (cf. A078147).

Adding n to each term a(n) gives A120327.
Positions of 0 are A013929.
Positions of 1 are A373415.
Positions of 2 are A378458.
Positions of 3 are A007675.
Sequences obtained by adding n to each term are placed in parentheses below.
The version for primes is A007920 (A007918).
The version for perfect powers is A074984 (A377468).
The version for squarefree numbers is A081221 (A067535).
The version for non-perfect powers is A378357 (A378358).
The version for prime powers is A378370 (A000015).
The version for non prime powers is A378371 (A378372).
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A120992 gives run-lengths of squarefree numbers increasing by one.
Cf. A073247, A151800, A236575, A243348, A375707, A377046, A378033, A378039, A378086, A378373.

Table[NestWhile[#+1&,n,SquareFreeQ[#]&]-n,{n,100}]

A378619 Distance between n and the greatest squarefree number <= n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 12 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..30000

Positions of 0 are A005117.
Positions of first appearances are A020755 - 1.
Positions of 1 are A053806.
Subtracting each term from n gives A070321.
The opposite version is A081221.
Restriction to the primes is A240473, opposite A240474.
A013929 lists the nonsquarefree numbers, differences A078147.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
Cf. A007674, A013928, A053797, A072284, A073247, A076259, A280892.
Cf. A007947, A067535, A081217, A112925.

Table[n-NestWhile[#-1&,n,!SquareFreeQ[#]&],{n,100}]

A378619(n) = forstep(k=n,1,-1,if(issquarefree(k), return(n-k))); \\ Antti Karttunen, Jan 29 2025

from itertools import count
from sympy import factorint
def A378619(n): return n-next(m for m in count(n,-1) if max(factorint(m).values(),default=0)<=1) # Chai Wah Wu, Dec 14 2024

a(n) = n - A070321(n).

Data section extended to a(105) by Antti Karttunen, Jan 29 2025

A378458 Squarefree numbers k such that k + 1 is squarefree but k + 2 is not.

Original entry on oeis.org

2, 6, 10, 14, 22, 30, 34, 38, 42, 46, 58, 61, 66, 70, 73, 78, 82, 86, 94, 102, 106, 110, 114, 118, 122, 130, 133, 138, 142, 145, 154, 158, 166, 173, 178, 182, 186, 190, 194, 202, 205, 210, 214, 218, 222, 226, 230, 238, 246, 254, 258, 262, 266, 273, 277, 282

Offset: 1

Gus Wiseman, Dec 02 2024

nonn

These are the positions of 2 in A378369 (difference between n and the next nonsquarefree number).

The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^2) - Product_{p prime} (1 - 3/p^2) = A065474 - A206256 = 0.19714711803343537224... . - Amiram Eldar, Dec 03 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A007675 within A007674.
The version for prime power instead of nonsquarefree is a subset of A006549.
Another variation is A073247.
The version for nonprime instead of squarefree is A179384.
Positions of 0 in A378369 are A013929.
Positions of 1 in A378369 are A373415.
Positions of 2 in A378369 are A378458 (this).
Positions of 3 in A378369 are A007675.
A000961 lists the powers of primes, differences A057820.
A120327 gives the least nonsquarefree number >= n.
A378373 counts composite numbers between nonsquarefree numbers.
Cf. A065474, A067535, A081221, A206256, A236575, A377046, A378033, A378039, A378086.

Select[Range[100],NestWhile[#+1&,#,SquareFreeQ[#]&]==#+2&]

list(lim) = my(q1 = 1, q2 = 1, q3); for(k = 3, lim, q3 = issquarefree(k); if(q1 && q2 &&!q3, print1(k-2, ", ")); q1 = q2; q2 = q3); \\ Amiram Eldar, Dec 03 2024

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A378369 Distance between n and the least nonsquarefree number >= n.

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A378619 Distance between n and the greatest squarefree number <= n.

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A378458 Squarefree numbers k such that k + 1 is squarefree but k + 2 is not.

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