A080733 -id:A080733 - cp's OEIS frontend

A081221 Number of consecutive numbers >= n having at least one square divisor > 1.

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

Offset: 1

Reinhard Zumkeller, Mar 10 2003

nonn

The first time terms 0..7 occur is at n = 1, 4, 8, 48, 242, 844, 22020, 217070. - Antti Karttunen, Sep 22 2017

			For n = 3, 3 is a squarefree number, thus a(3) = 0.
For n = 48, neither 48 = 2^4 * 3 nor 49 = 7^2, nor 50 = 2^2 * 5 are squarefree, but 51 = 3*17 is, thus a(48) = 3. - _Antti Karttunen_, Sep 22 2017

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A005117, A008683, A045882, A051681, A068781, A067535, A080733.

Flatten@ Map[If[First@ # == 0, #, Reverse@ Range@ Length@ #] &, SplitBy[Table[DivisorSum[n, 1 &, And[# > 1, IntegerQ@ Sqrt@ #] &], {n, 120}], # > 0 &]] (* Michael De Vlieger, Sep 22 2017 *)

A081221(n) = { my(k=0); while(!issquarefree(n+k),k++); k; }; \\ Antti Karttunen, Sep 22 2017

from itertools import count
from sympy import factorint
def A081221(n): return next(m for m in count(0) if max(factorint(n+m).values(),default=0)<=1) # Chai Wah Wu, Dec 04 2024

mu(k) = 0 for n <= k < n+a(n) and mu(n+a(n)) <> 0, where mu = A008683 (Moebius function).
a(n)*mu(n) = 0.
a(A068781(n)) > 0.
a(n) = A067535(n) - n. - Amiram Eldar, Oct 10 2023

A076260 a(n) = 0 if n is a squarefree number, otherwise the distance between the two nearest squarefree numbers around n: A067535(n)-A070321(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 3, 3, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 3, 3, 0, 3, 3, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 3, 3, 0, 0, 4, 4, 4, 0, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 3, 3, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 3, 3, 0, 0, 0, 3, 3, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 4, 4, 4, 0, 0, 0, 2, 0

Offset: 1

Reinhard Zumkeller, Oct 03 2002

nonn

a(n)=0 iff n is squarefree; otherwise a(n) > 1.

			The nearest squarefree numbers surrounding 25 = 5^2 are A070321(25) = 23 and A067535(25) = 26, therefore a(25) = 26-23 = 3. - Edited by _Antti Karttunen_, Nov 23 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005117, A020753, A020754, A076259, A080733.

Block[{nn = 105, s}, s = Select[Range[nn + 15], SquareFreeQ]; Array[If[FreeQ[s, #], First@ Differences@ s[[# - 1 ;; #]] &@ FirstPosition[Union@ Append[s, #], #][[1]], 0] &, 105]] (* Michael De Vlieger, Nov 23 2017 *)

A067535(n) = { while(!issquarefree(n), n++); n; } \\ These two functions from Michel Marcus, Mar 18 2017
A070321(n) = { while(!issquarefree(n), n--); n; }
A076260(n) = (A067535(n)-A070321(n)); \\ Antti Karttunen, Nov 22 2017

Definition corrected to match with the data as the old definition was that of A080733 - Antti Karttunen, Nov 23 2017

A087049 Characteristic sequence for numbers n>=0 that are either squares or have a square > 1 as factor.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1

Offset: 0

Wolfdieter Lang, Sep 08 2003

a(0)=1, a(1)=1, n>=2: a(n)=1 if isquarefree(n)=false else 0.
Except for a(0)=1 and a(1)=1 this is the bit-flipped unsigned Moebius sequence abs(A008683(n)), n>=2.
For n>=2: a(n)=1 iff n is from A013929 (not squarefree).

			a(4) = 1 because 4 is a square; a(8) = 1 because 8 = 2^2 * 2.

Cf. A008683, A008966, A080733, A000290 (squares), A013929 (not squarefree), A229099.

1,1,seq(`if`(numtheory:-issqrfree(n),0,1),n=2..100); # Robert Israel, Nov 17 2017

Array[If[# <= 1, 1, 1 - Abs@ MoebiusMu@ #] &, 105, 0] (* Michael De Vlieger, Nov 17 2017 *)

A087049(n) = if(n<=1,1,1-abs(moebius(n))); \\ Antti Karttunen, Nov 17 2017

a(n) = 1 if n is a perfect square (A000290) or has some square > 1 as a factor, else 0.
a(0) = a(1) = 1; for n > 1, a(n) = 1 - A008966(n). - Antti Karttunen, Nov 17 2017
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 1 - 6/Pi^2 (A229099). - Amiram Eldar, Jan 19 2024

A369164 a(n) = A001221(A000688(n)).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jan 15 2024

First differs from A369165 at n = 36, from A080733 at n = 49, and from A107078 at n = 72.

The sums of the first 10^k terms, for k = 1, 2, ..., are 3, 40, 426, 4307, 43203, 432211, 4322486, 43226028, 432261887, 4322622387, ... . From these values the asymptotic mean of this sequence, whose existence was proven by Ivić (1983) (see the Formula section), can be empirically evaluated by 0.43226... .

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter V, page 164.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Aleksandar Ivić, On the number of abelian groups of a given order and on certain related multiplicative functions, Journal of Number Theory, Vol. 16, No. 1 (1983), pp. 119-137. See p. 131, eq. 4.4.

Cf. A001221, A000688, A080733, A107078.
Cf. A048109, A059956, A271971.
Cf. A369162, A369163, A369165.

Table[PrimeNu[FiniteAbelianGroupCount[n]], {n, 1, 100}]

a(n) = omega(vecprod(apply(numbpart, factor(n)[, 2])));

Sum_{k=1..n} a(k) = c * n + O(sqrt(n) * log(n)^3/log(log(n))^2), where c = Sum_{k>=1} d(k) * A001221(k) is a constant, d(k) is the asymptotic density of the set {m | A000688(m) = k} (e.g., d(1) = A059956, d(2) = A271971, d(3) appears in A048109) (Ivić, 1983).

A309913 Distance from n to closest squarefree number that is different from n.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Aug 22 2019

nonn

Index entries for sequences related to distance to nearest element of some set

Cf. A005117, A051700, A080733, A316190.

a[n_] := Module[{k = 1}, While[! SquareFreeQ[n + k] && ! SquareFreeQ[n - k], k++]; k]; Table[a[n], {n, 0, 100}]

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A081221 Number of consecutive numbers >= n having at least one square divisor > 1.

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A076260 a(n) = 0 if n is a squarefree number, otherwise the distance between the two nearest squarefree numbers around n: A067535(n)-A070321(n).

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A087049 Characteristic sequence for numbers n>=0 that are either squares or have a square > 1 as factor.

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A369164 a(n) = A001221(A000688(n)).

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A309913 Distance from n to closest squarefree number that is different from n.

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