A053165 4th-power-free part of n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 2, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 3, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Author

Henry Bottomley, Feb 29 2000

Keywords

• nonn
• mult

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• Henry Bottomley, Some Smarandache-type multiplicative sequences.

Crossrefs

Equivalent sequences for other powers: A007913 (2), A050985 (3).

Cf. A000190, A000203, A008835.

A003961, A059897 are used to express relationship between terms of this sequence.

Related to A065331 via A225546.

Programs

Mathematica

f[p_, e_] := p^Mod[e, 4]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 07 2020 *)

PARI

a(n)=my(f=factor(n)); f[,2]=f[,2]%4; factorback(f) \\ Charles R Greathouse IV, Sep 02 2015

Python

from operator import mul
from functools import reduce
from sympy import factorint
def A053165(n):
    return 1 if n <=1 else reduce(mul,[p**(e % 4) for p,e in factorint(n).items()])
# Chai Wah Wu, Feb 04 2015

Formula

a(n) = n/A008835(n).

Dirichlet g.f.: zeta(4s)*zeta(s-1)/zeta(4s-4). The Dirichlet convolution of this sequence with A008835 is A000203. - R. J. Mathar, Apr 05 2011

From Peter Munn, Jan 15 2020: (Start)

a(2) = 2; a(4) = 4; a(n^4) = 1; a(A003961(n)) = A003961(a(n)); a(A059897(n,k)) = A059897(a(n), a(k)).

a(A225546(n)) = A225546(A065331(n)).

(End)

Multiplicative with a(p^e) = p^(e mod 4). - Amiram Eldar, Sep 07 2020

Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / 210. - Vaclav Kotesovec, Aug 20 2021