A087049 -id:A087049 - cp's OEIS frontend

A107078 Whether n has non-unitary prime divisors.

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, 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

Offset: 1

Paul Barry, May 10 2005

Also the characteristic function of the numbers that are not squarefree: A013929. - Enrique Pérez Herrero, Jul 08 2012

The sequence of partial sums of this sequence is A057627. - Jason Kimberley, Feb 01 2017

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Index entries for characteristic functions.

Cf. A087049. - R. J. Mathar, Aug 24 2008

Cf. A013929, A008683, A008966, A107078, A229099.

seq(1 - abs(numtheory:-mobius(n)), n = 1..101); # Peter Luschny, Jul 27 2023

Table[1-MoebiusMu[n]^2,{n,1,100}] (* Enrique Pérez Herrero, Jul 08 2012 *)

from sympy import mobius
def A107078(n): return int(not mobius(n)) # Chai Wah Wu, Dec 05 2024

a(n) = 1 if A056170(n)>0, 0 otherwise.
a(n) = A107079(n) - A013928(n+1).
a(n) = 1 - A008966(n). - Reinhard Zumkeller, Oct 03 2008
a(n) = Sum_{k=0..n-1} (mu(n-k-1) mod 2) - Sum_{k=0..n-1} (mu(n-k) mod 2).
a(n) = abs(mu(n) - (-1)^omega(n)) = (mu(n) - (-1)^omega(n))^2 = abs(A008683(n) - (-1)^A001221(n)). - Enrique Pérez Herrero, Apr 28 2012
a(n) = 1 - mu(n)^2. - Enrique Pérez Herrero, Jul 08 2012
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 - 6/Pi^2 (A229099). - Amiram Eldar, Jul 24 2022

A096309 a(1)=1; for n > 1, a(n) is the number of levels in the "stacked" prime number factorization of n (prime number factorization of the exponents if necessary and so on ...).

Original entry on oeis.org

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

Offset: 1

Franz Vrabec, Jun 27 2004

For n > 1: a(n)=1 iff n squarefree.

Sequence A185102 is a (better?) variant, identical except for A185102(1)=0. - M. F. Hasler, Nov 21 2013

			a(4)=2 because 4=2^2; a(8)=2 because 8=2^3; a(16)=3 because 16=2^(2^2).
a(65536) = a(2^2^2^2) = a(2^^4) = 4 is the first term larger than 3; the index of the first a(n) > 4, n = 2^^5, has 19729 digits. - _M. F. Hasler_, Nov 21 2013

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

Cf. A087049, A185102.

f[n_Integer] := FactorInteger[n][[All, 2]]; a[n_] := Depth[f[n] //. k_Integer /; k > 1 :> f[k]] - 1; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 20 2013 *)

A096309=n->if(n>1,vecmax(apply(a,factor(n)[,2])))+1 \\ M. F. Hasler, Nov 21 2013

More terms from Jean-François Alcover, Nov 20 2013

A087050 Square root of the largest square >1 dividing the n-th nonsquarefree number.

Original entry on oeis.org

2, 2, 3, 2, 4, 3, 2, 2, 5, 3, 2, 4, 6, 2, 2, 3, 4, 7, 5, 2, 3, 2, 2, 3, 8, 2, 6, 5, 2, 4, 9, 2, 2, 3, 2, 4, 7, 3, 10, 2, 6, 4, 2, 3, 2, 11, 2, 5, 3, 8, 2, 3, 2, 2, 12, 7, 2, 5, 2, 3, 2, 4, 9, 2, 2, 13, 3, 2, 5, 4, 6, 2, 2, 3, 8, 14, 3, 10, 2, 3, 4, 2, 6, 2, 4, 15, 2, 2, 3, 2, 4, 11, 9, 2, 7, 2, 5, 6, 16, 2, 3

Offset: 1

Wolfdieter Lang, Sep 08 2003

			n=10, A013929(10) = 27, a(10)^2 = 3^2 = 9. 27 = 9*3.
n=39, A013929(39) = 100, a(39)^2 = 10^2 = 100.

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

Cf. A087049, A013929, A000188, A000290.

Cf. A001610, A013661, A306016.

f[p_, e_] := p^Floor[e/2]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; s /@ Select[Range[300], !SquareFreeQ[#] &] (* Amiram Eldar, Feb 11 2021 *)

from math import isqrt, prod
from sympy import mobius, factorint
def A087050(n):
    def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return prod(p**(e>>1) for p, e in factorint(m).items() if e>1) # Chai Wah Wu, Jul 22 2024

a(n)^2 is the largest square factor (from A000290) of the nonsquarefree number A013929(n), n>=1.
a(n) = A000188(A013929(n)). - Amiram Eldar, Feb 11 2021
Sum_{k=1..n} a(k) ~ (n/(2*(zeta(2)-1))) * (log(n) + 3*gamma - 3 - 2*zeta'(2)/zeta(2) - log(1-1/zeta(2))), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 14 2024

cp's OEIS Frontend

A107078 Whether n has non-unitary prime divisors.

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A096309 a(1)=1; for n > 1, a(n) is the number of levels in the "stacked" prime number factorization of n (prime number factorization of the exponents if necessary and so on ...).

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A087050 Square root of the largest square >1 dividing the n-th nonsquarefree number.

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