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

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

Author

Gus Wiseman, Dec 12 2024

Keywords

• nonn

Links

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

Crossrefs

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.

Programs

Mathematica

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

PARI

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

Python

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

Formula

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

Extensions

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