

A053165


4thpowerfree part of n.


11



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
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OFFSET

1,2


LINKS

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


FORMULA

a(n) = n/A008835(n).
Dirichlet g.f.: zeta(4s)*zeta(s1)/zeta(4s4). 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


MATHEMATICA

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


PROG

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


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.
Sequence in context: A002377 A053836 A025483 * A056962 A043275 A216455
Adjacent sequences: A053162 A053163 A053164 * A053166 A053167 A053168


KEYWORD

nonn,mult


AUTHOR

Henry Bottomley, Feb 29 2000


STATUS

approved



