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A053165 4th-power-free part of n. 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
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
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).
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
KEYWORD
nonn,mult
AUTHOR
Henry Bottomley, Feb 29 2000
STATUS
approved

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Last modified March 29 05:16 EDT 2024. Contains 371264 sequences. (Running on oeis4.)