A240473 -id:A240473 - cp's OEIS frontend

A240476 Primes that are not midway between the closest flanking squarefree numbers.

3, 5, 7, 11, 13, 23, 29, 31, 37, 41, 43, 47, 59, 61, 67, 71, 73, 79, 83, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 167, 173, 179, 181, 191, 193, 211, 223, 227, 229, 239, 241, 257, 263, 277, 281, 283, 311, 313, 317, 331, 347, 349, 353

Offset: 1

Chris Boyd, Apr 06 2014

nonn

Primes for which the corresponding A240473(m) is not equal to A240474(m).
Primes not equal to the average of the closest flanking squarefree numbers.
Primes not equal to the average of three consecutive squarefree numbers.
Complement of A240475 relative to A000040.

			29 is a term because it is not midway between the closest flanking squarefree numbers 26 and 30.
On the other hand, 19 is not a term because it is midway between the closest flanking squarefree numbers 17 and 21.

Chris Boyd, Table of n, a(n) for n = 1..10000

Cf. A000040, A075430, A075432, A240473, A240474, A240475

forprime(p=1,353,forstep(j=p-1,1,-1,if(issquarefree(j),L=j;break));for(j=p+1,2*p,if(issquarefree(j),G=j;break));if(G-p!=p-L,print1(p", ")))

A318959 Primes p (> 2) such that p - 2 and p - 1 are nonsquarefree.

Original entry on oeis.org

29, 101, 127, 137, 149, 173, 277, 281, 317, 353, 389, 461, 509, 541, 569, 577, 641, 677, 727, 821, 857, 877, 929, 977, 1109, 1129, 1181, 1217, 1277, 1289, 1361, 1423, 1433, 1451, 1613, 1667, 1721, 1777, 1861, 1877, 1901, 1913, 1973, 2081, 2153, 2297, 2333, 2351

Offset: 1

Seiichi Manyama, Sep 06 2018

nonn

			21 (= 23 - 2) is squarefree. So 23 is not a term.
27 = 3^3 and 28 = 2^2*7. So 29 is a term.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A039787, A049231, A240473, A257545, A319049.

[p: p in PrimesInInterval(3, 2500)| not IsSquarefree(p-2) and not IsSquarefree(p-1)]; // Vincenzo Librandi, Sep 06 2018

Select[Prime[Range[500]], !SquareFreeQ[# - 2] && !SquareFreeQ[# - 1] &] (* Vincenzo Librandi, Sep 06 2018 *)

forprime(p=2, 1e4, if(!issquarefree(p-1)&&!issquarefree(p-2), print1(p, ", "))); \\ Altug Alkan, Sep 06 2018

A319049 Primes p such that none of p - 1, p - 2 and p - 3 are squarefree.

Original entry on oeis.org

101, 127, 353, 727, 1277, 1423, 1451, 1667, 2153, 2351, 2647, 3187, 3251, 3511, 3701, 3719, 3727, 4421, 4951, 5051, 5393, 5527, 6427, 6653, 6959, 7517, 7867, 8527, 9127, 9551, 9803, 9851, 10243, 10253, 10487, 10831, 11273, 11351, 11777, 11827, 12007, 12251, 12277

Offset: 1

Seiichi Manyama, Sep 08 2018

nonn

If p is a term, so that there are primes q,r,s such that q^2|p-3, r^2|p-2 and s^2|p-1, then the sequence includes all primes == p (mod q^2*r^2*s^2). In particular, the sequence is infinite, and a(n)/(n*log(n)) is bounded above and below by constants. - Robert Israel, Sep 09 2018

			98 = 2*7^2, 99 = 3^2*11 and 100 = 2^2*5^2. So 101 is a term.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A039787, A049231, A240473, A257545, A318959.

[p: p in PrimesUpTo(13000) | not IsSquarefree(p-1) and not IsSquarefree(p-2) and not IsSquarefree(p-3)]; // Vincenzo Librandi, Sep 17 2018

Res:= NULL: count:= 0:
p:= 1;
while count < 100 do
  p:= nextprime(p);
  if not ormap(numtheory:-issqrfree, [p-1,p-2,p-3]) then
    count:= count+1; Res:= Res, p
  fi
od:
Res; # Robert Israel, Sep 09 2018

Select[Prime[Range[2000]], !SquareFreeQ[# - 1] && !SquareFreeQ[# - 2] && !SquareFreeQ[# - 3]&] (* Jean-François Alcover, Sep 17 2018 *)
Select[Prime[Range[1500]],NoneTrue[#-{1,2,3},SquareFreeQ]&] (* Harvey P. Dale, Apr 11 2022 *)

isok(p) = isprime(p) && !issquarefree(p-1) && !issquarefree(p-2) && !issquarefree(p-3); \\ Michel Marcus, Sep 09 2018

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

cp's OEIS Frontend

A240476 Primes that are not midway between the closest flanking squarefree numbers.

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A318959 Primes p (> 2) such that p - 2 and p - 1 are nonsquarefree.

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A319049 Primes p such that none of p - 1, p - 2 and p - 3 are squarefree.

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

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